In his article "On the Foundations of Geometry", Poincaré goes through an extensive discussion to establish that our experience of motion may be properly regarded as displacements. This is requisite to proving that our experience of the world is characteristic of groups (in the mathematical sense of the term). He does this by arguing that an external change A can be undone by an voluntary change A'. For example, if an object moves from the left side to the right of my field of vision, I can voluntarily restore it to its original position on the left by redirecting the direction of my gaze. Proceeding with this line of thought, he goes on to assert the following:
Nor is this all. If two external changes α and α' are regarded as identical on the basis of the convention adopted above, or in other words, are susceptible of being corrected by the same internal change A; if, on the other hand, two other external changes β and β' can be corrected by the same internal change B, and consequently may also be regarded as identical, have we the right to conclude that the two changes α + β and α' + β' are susceptible of being corrected by the same internal change, and are consequently identical? Such a proposition is in no wise evident, and if it be true it cannot be the result of a priori reasoning.
His argument seems to betray a misunderstanding of the issue. If these relations are determined a priori, questioning whether or not they are immediately evident seems to be rather idle. What needs to be asked is whether it's possible for us to implement these internal changes, B + A, which would result in correcting, let's say, α + β, but having some entirely different result from the expected correction of α' + β'. Similarly, it might be asked if it could result in correcting α' + β' but having an unexpected result with respect to α + β.
The nature of a priori truth has nothing to do with how easily it may be verified; rather, it pertains to whether it is determined by our faculties as opposed to being determined by the world represented by those faculties. Kant makes this point clear by pointing out that the understanding has its own particular nature which may not be immediately evident to us:
"[The categories] are merely rules for an understanding... But to show reasons for this peculiar character of our understandings, that it produces unity of apperception a priori only by means of categories, and a certain kind and number thereof, is as impossible as to explain why we are endowed with precisely so many functions of judgement and no more, or why time and space are the only forms of our intuition."
Did Poincaré simply not understand the nature of a priori truth, or is there some other argument that he employs to establish his claim, i.e. an argument that has nothing to do with whether or not things are immediately evident?