Perhaps the biggest group of examples of people thinking that they are not politically correct when they actually are) is when they argue in favour of impartiality/ neutrality of procedures.
I will call such a person 'would-be anti-PC' - they want to be anti-PC but are trapped into supporting politcal correctness by the unavoidably-PC nature of modern discourse.
The would-be anti-PC are, in fact, trapped into supporting PC by their desire to 'make a difference', by their desire to engage with the modern socio-political-managerial discourse, by their wish to appeal as widely as possible and to build a 'movement'.
High level (roughly 'managerial') modern discourse only permits variants of PC arguments, such that people who wish to oppose PC either have no effect, or lose their positions, or actually strengthen PC.
For example, if someone opposes 'diversity'/ 'affirmative action'/ group preferences for college admissions then it seems almost inevitable that his argument be made in terms of wanting (for instance) a 'sex blind' system.
And in a sense that is almost-exactly what he does want, the would-be anti-PC wants sex not to be an important feature in college admissions, wants other factors (educational, cultural, moral etc) to predominate.
So a person who opposes mantras such as 'we ought to appoint a woman', and opposes the cyclical celebrations of appointments being made to 'a woman' will argue (usually, indeed almost invariably) that it 'shouldn't matter' about the sex of the appointee.
But this is a mistake at multiple levels.
In the first place, as we should (by now) know from experience, arguments for neutrality are utterly ineffective against PC.
They are, however, acceptable within PC.
Arguments in favour of procedural neutrality are acceptable precisely because they are utterly ineffective!
(If such neutral arguments happen to become effective at some point - for example in Ward Connerly's anti-AA initiatives - then the neutrality argument become demonized by PC precisely as if it was not neutral but was advocating the opposite of PC-approved affirmative action. Since PC intervenes to reverse the natural and spontaneous effect of neutral systems, and judges by outcomes not processes, then for PC you must discriminate; and the only substantive debate is who are the favoured groups.)
Yet the sex of the appointee does matter, that is the whole reason for the debate in the first place.
For proponents of procedural neutrality to say that it 'shouldn't' matter is already to be creating an abstract ideal world of pure impersonal functionality - a world in which the 'efficiency' of systems (efficiency being inevitably measured on the basis of a selected criteria over a limited time frame, and with no reason to assume the measure is generalizable to other variable and other time frames)... in which efficiency of systems is intrinsically more important than the people in the systems and the people who are affected by the systems.
Procedural neutrality therefore contradicts spontaneous common sense just as PC contradicts common sense ; and in this specific domain PC is in fact closer to common sense (in its assumption that men and women are different and need different treatment) than is procedural neutrality, which is presumably why PC wins most of these debates.
(Another example: Common sense says that bad people are more likely to do bad things than good people, so the law should take character into account in determining guilt. But specialized procedural modernity has it that guilt must be established by objective facts that take no account of personality.
(At the end of this process of abstraction, schoolteachers find that they are unable to take account of character in determining guilt: a well behaved and habitually truthful child accuses a badly behaved and dishonest child of theft; the bad child denies theft; lacking objective evidence (the stolen money has been spent) the situation is framed as 'one person's word against another' and the bad child goes undetected and unpunished.)
(Multiply this situation by ten million and you have modern justice. )
(The situation arises because the PC advocate realizes that the good child might for once, be lying. And that, since humans are sinful and corruptible, the good child might for once be dishonest. Worse, the good child has an incentive to be dishonest. And that any system which believes good people and disbelieves bad people is an oligarchy of the good over the bad (yet nobody is wholly good, nor wholly bad); and the good will also tend to become corrupted and (in principle) good may become bad and bad good.)
(All of these criticisms are true - but all of these are taken to be reasons why abstract systems are preferable to common sense - which is merely an assumption unless you believe that it is possible for an abstract system to be free of human sinfulness; and that humans would know when this was the case; and would adopt, implement and sustain the sin-free abstract system. To believe this is pure unfounded assumption with zero evidence in favour and vast shoals of evidence against. However all of these things are indeed believed by PC, which is why PC is psychotic - in the sense of being based on absurd premises.)
We cannot pretend that a system of men and a system of women is or should be seen as 'the same' - except in a very narrow, selective, partial and biased sense of 'the same' which begs all the important questions.
In effect, to say that men and women are 'the same' (or the same in practice) is to make an assumption - not to report an empirical situation.
Modern systems have not discovered that men and women are the same - they have assumed it.
Empirical evidence, as usual, cannot settle the question (although almost all evidence shows differences between groups of men and women - this is never enough, and can always be reinterpreted to favour either side).
Neutrality about important matters (such as whether a position is held by a man or a woman) is in the first place impossible, and in the second place actually undesirable as a goal even if it were not impossible - although the problems of neutrality typically do not emerge quickly.
(Indeed, and this is Charlton's Law - really just a generalization, rather than a law - i.e. that all actually-enacted policies that are harmful overall in the long term, are specifically advantageous in the short term - which is why they get adopted. And since specific advantage is easier to define and measure, the fact of overall harmfulness is not effective at getting the policy removed or reversed. E.g. a policy which benefits a specific interest group - e.g. subsidies to mohair farmers - can survive for generations despite causing overall harm.)
My contention is that the West took a wrong turn about these matters around AD 1000 (the Great Schism) when the Roman Catholic church (gradually) broke away from the Eastern Orthodox - and like most mistakes it was initially richly rewarded (otherwise the mistake would not be made) - by tremendous 'progress' in philosophy and scholarship (especially Thomas Aquinas), then in science, and later in the economy.
The mistake was built-in specialization; progressive specialization of all functions, with no end or limit.
And the Great Schism built-into the thought systems of the West a fatal error, of which PC is a remote and indirect consequence.
By secularizing knowledge, by creating The University - autonomous philosophy (instead of having learning focused in monasteries) the West eventually made political correctness - which is now everywhere and inescapable in the West.
(My contention is that PC is not a sideshow, but the main feature in the West.)
And PC is the West's Nemesis, because the West cannot decisively overturn PC without overturning that which made it The West.
The West cannot overcome PC without ceasing to be The West.
Yet, if this overturning of PC does not happen, then the West will itself be overturned.
This is not a paradox: it is an outline of possible futures.
And the delusion of being non-PC is widespread ultimately because to be anti-PC is to be anti-West (in tendency, albeit not in intention).
Yet at the same time, to be pro-PC is (obviously) to be anti-West (sometimes in intention, but always in tendency).
But then The West never was sustainable, it was a blip (albeit a thousand year 'blip'!).
And as soon as The West began to implement its assumptions towards completion (which is political correctness), just so soon The West began actively (as well as passively) to destroy itself.
NOTE: The apparent exception is those who advocate orthodox Roman Catholicism as a theocratic principle. By the exception is only apparent, in my opinion.
My critique is that RC is semi-modern, hence has never been able to hold the line against ever more progressive functional specialization.
Aquinas created the 'perfect' philosophy - but it was almost immediately swept-away (by Scotus, Occam etc) - since when philosophical change has been almost continuous, generation upon generation, and attempts to hold the line with Thomism have never succeeded - never remotiely succeeded.
Since the philosophical base of RC has continually been changing, RC has continually been changing - modernizing: become ever more functionally specialized.
This is because RC includes decisive features of modernity: independence of philosophy from theology, universities from monasteries, the Pope from tradition - in a nutshell, ideology from holiness...
And once modernity (progressive specialization) has been built into a system, it is cannot stop itself - and it cannot stop itself because continual specialization creates continual autonomy, so there is no way of one part of the system stopping another.
You just get more and more specialization until the whole social system falls apart into dysfunctionally small fragments; and all the King's Horses and all the King's men cannot put Humpty together again.