Exactly. The image in a mirror is physically symmetrical to the direct image from objects in the real world, with the mirror itself as the plane of symmetry.

I find the front-to-back explanation to give a better intuition than the typical "but left and right are relative to your body", which only serves to explain the paradox of "left-right are reversed but up-down ar not", but not the fact that everything looks flipped.

Well, it flips front-to-back. But this side of the mirror, we can't physically do that, so it's not in our brain's list of expected possibilities. We can, however, rotate ourselves 180 degrees. And because we're bilaterally symmetrical, our brain finds it easy to interpret the front-to-back flip as a 180 degree rotation. But that leaves left-to-right weird, because what happened wasn't actually a 180 degree rotation.