[mdbco]

It's actually entirely possible to do some "checking of your answer" for p-values, as well. As you mentioned, for practical math, you can often validate your answer against the initial assumptions. This is true for statistical testing too, as it typically relies on many theoretical assumptions. So what you can do in practice is propose a different set of assumptions, perform the hypothesis test in a manner that follows those new assumptions, and see if you obtain a similar result. Typically, for any given hypothesis that you want to test, there are several possible methods for performing that test, so you can redo your test many times. This is one type of robustness checking, which includes many other things as well (e.g. running your test over subsamples or resamples of the data, checking for sensitivity to outliers, etc). Good statisticians generally like to do lots and lots of robustness checking.