One of the things the past few years has made clear to me is that there are many, many supposedly intelligent people who nonetheless cannot grasp the elementary point that 'centrism' is a miscellaneous category; it is just the term we use in political matters for anything that doesn't definitely fit into 'left' or 'right'. This should be extraordinarily easy to figure out. 'Centrism' itself is a relative label (center compared to what?), designating what is neither definitely to the right or definitely to the left. It is why there are so very many people who count as centrists of some kind but almost all politics in modern liberal societies gets divided according to some local version of a left vs. right divide -- there is no stable position or set of positions that is 'centrism', even locally, but lots of different leftover positions. And centrists who self-identify as centrists always do so because they don't think that they count as typical 'left' or 'right'. 'Centrism' is capable of covering any number of very different positions.
It's an interesting question why this seems to be so difficult for people, even people who specialize in fields like political philosophy, political science, or history, where you would expect them at least to ask the question of how people get sorted into the category to begin with. I suspect it comes down to two things:
(1) Left-vs.-right talk makes these sound like substantive and stable options along a line. In reality, 'left' and 'right' are none of these things. Even the historical reason for talking about 'left' and 'right' has nothing to do with a line; it seems to trace to a historical accident about which side of the legislative chamber different factions happened to sit on at some point. The terminology stuck, and the reason for assigning people to 'left' or 'right' is just based on a crude sense of precedent -- those people are 'left' who seem to us most like the 'left' of the previous generation, and those people are 'right' who seem to us most like the 'right' of the previous generation. And this means that it wavers all over the place on particular details. Easy movement across borders, for instance, was a 'right' idea opposed by the 'left' that became a 'left' idea opposed by the 'right'. It's just the overall package that gives the labels. And 'left' and 'right' vary considerably according to society. The 'right' in Canada is very different from the 'right' in America.
In reality, politics has many dimensions. When we get to more serious analysis, this becomes obvious -- the alliance of certain fiscal and social positions is obviously due to historical accident, which is why primarily-fiscal and primarily-social factions of both the 'left' and the 'right' rarely get along very well. There is no obvious reason why someone who is 'left' on labor issues should always be 'left' on immigration issues, no obvious reason why being 'right' on marriage must always go with being 'right' on tax cuts. People get sorted into two groups just because binary contests are easier to follow than complicated ones (there are incentives for lumping as many of your opponents into a single group if possible, so that you have an Us faction and a Them faction), and because self-identification as 'left' and 'right' indicates a willingness to ally -- even with gritted teeth -- with other groups who already self-identify as such.
But the fact that we do sort people this way, and the fact that we can pretty easily identify some of the alliances at any given point of time, means that the way we use 'left' and 'right' makes it sound like these are definite positions along one and only one line. And if they were, then 'centrism' would presumably be the definite center of the line.
(2) The name tends toward a confusion of 'moderate' with 'centrist'. This is something you even find in dictionaries. I recently saw a tweet that said that 'centrism' was a false application of the Doctrine of the Mean to politics. This is an immensely stupid thing to say; Aristotle's account of the Doctrine of the Mean directly says that the moderate or mean is not the central point, because what counts as moderate depends on what the extremes are, and the real mean is typically closer to one of the extremes than the other. In addition, the claim requires assuming that left and right are unitary and the only real, stable, and definite directions in politics; this is entirely false. And it is simply not the general motivation for centrism. The most common motivation for centrism -- or at least the most significant in the history of centrisms -- is that people started on the 'left' or 'right', but something happened that made them unwilling to identify as definite allies of whichever one they were, without giving them reason to identify as definite allies of the other side. Alienation, not moderation, is their primary and explicit reason. (Indeed, one of the most widely recognized experiences in contemporary politics in liberal societies is how thoroughly alienating disputes between 'left' and 'right' are; people complain about it all the time. The sense of having no real place in the standard political discussion is a widespread result of the way modern societies work.) Perhaps another common motivation is a mix-and-match of different positions, so that on any one issue they would be classifiable as 'left' and 'right', but not always or even usually the same. There is nothing about the classification that makes this impossible; you could only rule it out if you thought, again, that 'left' and 'right' were substantive and stable options giving direction to a single line.
In reality, of course, everybody everywhere should usually be moderate according to the reasoning of a reasonable person; while probably very few are, I suppose most people at least assume that they are so, and most of the rest are assuming that they have some reasonable emergency justification for extreme measures. Given this, it is inevitable that people on the 'left' and 'right' will often think that 'centrism' is gibberish -- an attempt at finding a middle ground between being reasonable and being unreasonable. But this is an imposed interpretation, and not a discovered fact.