EDIT: Let me make a few additional comments to nudge the discussion in a more technical and less opinion-based direction.

Suppose I don't buy the "argument from geometric intuition" and I seek a more algebraic (or combinatorial) explanation of the existence of the alternating group. It still seems to me that an argument based on determinants, or the distinction between $\SO(n)$ and $\O(n)$, would be a more "conceptual" route than an argument based on actions on the complete graph. For example, suppose I ask the corresponding question for the hyperoctahedral group ("why is there this subgroup of index 2?"). If we've already decided that the relationship between $\SO(n)$ and $\O(n)$ is the key point, then the same explanation works for both questions. Whereas, the graph-theoretic approach doesn't seem to have the same ability to give a unified explanation for both cases.

Am I mistaken about this? Does Cartier's proof generalize to cover all situations that I might try to explain by appealing to the distinction between $\SO(n)$ and $\O(n)$? Does it perhaps generalize even further to explain index-2 subgroups that don't seem to be explicable in terms of $\SO(n)$?

This question is prompted by a recent highly-upvoted question, Conceptual reason why the sign of a permutation is well-defined? The responses made me realize that my intuition differs from that of many other mathematicians, in a way that I had previously been unaware of.

As a bit of personal background, I recall being taught in middle school that (in not so many words) that there are "24 symmetries of a cube," since you can rotate any face to the bottom (a factor of 6) and then rotate that face in place (a factor of 4). Similarly, for each of the Platonic solids, we can count $4\times 3 = 12$ "symmetries" of a tetrahedron, $8\times 3 = 24$ "symmetries" of an octahedron, and so forth.

One of the various equivalent formulations of the aforementioned MO question is, "What is the conceptual explanation for the existence of the alternating group?" In part because of my background, the answer that came to my mind was, "Because it's the group of symmetries of a simplex." However, people rightly pointed out that to conform to standard usage, this answer should really be phrased, "Because it's the group of orientation-preserving symmetries of a simplex." But once the statement is phrased this way, it suggests that maybe the existence of the alternating group isn't such a "basic" fact after all; maybe it's only the symmetric group whose existence can be taken as basic, and we have to "explain" the existence of a subgroup of index 2 using some kind of algebraic argument.

There's no question that Poonen's version of Cartier's argument is slick and beautiful. Nevertheless, something doesn't quite sit right with me if we call this an "explanation" of the existence of the alternating group. It still seems to me that orientation-preserving rigid motions in Euclidean space are mathematically fundamental, because they are rooted in our physical intuition. Admittedly, in modern mathematics, formally capturing our physical intuition in a direct manner is a cumbersome process; we have to construct the real numbers, and then talk about continuous transformations that preserve the metric. But for me, the complexity of this formal definition does not imply that the underlying concept is complex; rather, it says more about the difficulties involved in formalizing our geometric intuition. It is not hard for me to imagine an alternative universe in which we begin with $SO(n)$, and think of $O(n)$ as an extension of $SO(n)$, much as Spin came before Pin.

But let me now finally come to my question. For those who take the symmetric group as given, and feel the need for an algebraic explanation of a subgroup of index 2, do you feel the same way about other finite subgroups of $SO(n)$? For example, do you think the existence of the hyperoctahedral group of order $2^n n!$ is "obvious," but the subgroup of index 2 is in need of an algebraic "explanation"? If so, is there an algebraic proof that you feel explains all these "miraculous" subgroups of index 2 in a uniform way? Or is the existence of orientability itself a miraculous fact?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$This question is prompted by a recent highly-upvoted question, Conceptual reason why the sign of a permutation is well-defined? The responses made me realize that my intuition differs from that of many other mathematicians, in a way that I had previously been unaware of.

As a bit of personal background, I recall being taught in middle school that (in not so many words) that there are "24 symmetries of a cube," since you can rotate any face to the bottom (a factor of 6) and then rotate that face in place (a factor of 4). Similarly, for each of the Platonic solids, we can count $4\times 3 = 12$ "symmetries" of a tetrahedron, $8\times 3 = 24$ "symmetries" of an octahedron, and so forth.

One of the various equivalent formulations of the aforementioned MO question is, "What is the conceptual explanation for the existence of the alternating group?" In part because of my background, the answer that came to my mind was, "Because it's the group of symmetries of a simplex." However, people rightly pointed out that to conform to standard usage, this answer should really be phrased, "Because it's the group of orientation-preserving symmetries of a simplex." But once the statement is phrased this way, it suggests that maybe the existence of the alternating group isn't such a "basic" fact after all; maybe it's only the symmetric group whose existence can be taken as basic, and we have to "explain" the existence of a subgroup of index 2 using some kind of algebraic argument.

There's no question that Poonen's version of Cartier's argument is slick and beautiful. Nevertheless, something doesn't quite sit right with me if we call this an "explanation" of the existence of the alternating group. It still seems to me that orientation-preserving rigid motions in Euclidean space are mathematically fundamental, because they are rooted in our physical intuition. Admittedly, in modern mathematics, formally capturing our physical intuition in a direct manner is a cumbersome process; we have to construct the real numbers, and then talk about continuous transformations that preserve the metric. But for me, the complexity of this formal definition does not imply that the underlying concept is complex; rather, it says more about the difficulties involved in formalizing our geometric intuition. It is not hard for me to imagine an alternative universe in which we begin with $\SO(n)$, and think of $\O(n)$ as an extension of $\SO(n)$, much as Spin came before Pin.

But let me now finally come to my question. For those who take the symmetric group as given, and feel the need for an algebraic explanation of a subgroup of index 2, do you feel the same way about other finite subgroups of $\SO(n)$? For example, do you think the existence of the hyperoctahedral group of order $2^n n!$ is "obvious," but the subgroup of index 2 is in need of an algebraic "explanation"? If so, is there an algebraic proof that you feel explains all these "miraculous" subgroups of index 2 in a uniform way? Or is the existence of orientability itself a miraculous fact?