Arranging your probability estimates so that predictions of opposite utility cancel out is one way to satisfy the anti-mugging axiom. ... There's no rule that says that similar statements with positive and negative utilities have to have the same prior probabilities, unless you introduce it specifically for the purpose of anti-mugging defense.

I believe there is such a rule, which doesn't have to be introduced ad hoc, and which follows from the tenets of algorithmic information theory. Per the reasoning I gave in the linked post, an arbitrary complex conclusion you locate (like the one in Pascal's mugging) necessarily has a corresponding conclusion of equal complexity, but with the right predicate(s) inverted so that the inferred utility is reversed.

Because (by assumption) the conclusion is reached through arbitrary reasoning, disentangled from any real-world observation, you need no additional complexity for a hypothesis that critically inverts the first one. Since no other evidence supports either conclusion, their probability weights are determined by their complexity, and are thus equal.

That's why I don't think you need to introduce this reasoning as an additional axiom. However, as a separate matter (and whether or not you need it as an axiom), I thought this argument was refuted by the fact that the mugger, simply through assertion, introduces an arbitrarily small amount of evidence favoring one hypothesis over its inverse. If it refutes the defense I gave in the link, it should work against the anti-mugging axiom you're using as well.