Just adding to the "a bit of both" chorus. I visualize and then (if still in more doubt than I'm comfortable with) count cubes and check angles in the supposedly matching bits. In some cases I count/check some parts but not others.

(I think my visualization skills are pretty weak, but I may be comparing myself with other mathematicians rather than the general population, which might well produce skewed results.)

I'm not really sure how to answer this question. I can do pretty well at these simply by visualizing the rotation, and in a casual setting, where not much hinged on getting it right, I would probably just use that method because it's a lot faster and easier. But consciously checking relations does improve my performance (or at least feels more reliable), so if I was in a context where it was important to get the answers right, that's what I would do.

It's like asking whether I add up three-digit numbers in my head or work it out on paper. Depends on the context.

When I need to be 95% sure I count the cubes (which is what I answered), but when being 70% sure suffices I visualize the rotation.

I visualise the rotations up until the point where it's too complex, after which I resort to checking relations

I chose "consciously check relations", but I'm nearly as bad at doing it that way as at attempting to visualise the rotation. I find these problems almost impossible. (I thiiiink the answers for the examples are a: same and b: different, but I'm a long way from completely sure: I think I'd have to build them with those little cubes they give primary school children in maths classes to be sure.) I guess it makes sense that people who are weaker at mental rotation (or, as other commenters suggest, want to be really sure of getting the right answer) resort to conscious checking, so if women on average do worse at mental rotation, you'd expect to find more conscious checkers among them.

Asking for the answers to (a) and (b) would have been worthwhile.

Like many people have said, I do it in my head except when I don't.

That said, having done a lot of neuropsych testing after recovering from my stroke, I've developed more confidence than I used to have in my mental rotation reliability; I'm more comfortable accepting a "yeah, that seems right" judgment without confirming it.

Interesting result from misreading the question: I first tried to resolve this as "are the first and second figures in (b) a rotation of the first and second figures in (a)?"

Trying to mentally rotate the first figure in (a) to get the first figure in (b) crashed the entire process, to the point where I began to doubt whether I could mentally rotate things in my mind. On the other hand, it was immediately clear to me on inspection that they weren't the same, in a similar way that it's clear an acute and an obtuse angle aren't the same angle.

However, resolving the question as intended produces the subjective experience of "rotating in my mind".

Your poll is somewhat broken (last option missing). Note that ability to rotate in the mind is very differently expressed. Some people do it effortlessly, some even with multiple elements (Tesla was said to be able to animate whole machines in his mind). Therefore I'd recommend to provide a scaled or indexical poll ("not at all", "partial/limited", "single element single rotation", "single element, multiple motions/changes", "multiple elements interacting (gears)", "whole machines"). As only 4 people (me included) voted I recommend to repost the poll and extend it.