I cannot tell what opinion the neuroscience holds here, i.e., if you do a brain scan of a person trying to read a mathematical text or solve a problem, and then compare it with the scan of a person trying to determine the shape of an object by touch, smelling a flower, or tasting some food or drink, which would be the common areas to light up. The experiment like that is not too hard to carry out on a few subjects from laymen to professional mathematicians and the results may be quite interesting, so I'm inclined to believe that somebody has carried it out already, but, alas, I have no idea who and when. What I describe below is just my personal experience.
The answer is "all and none". Let's talk about vision, even. I mainly think in pictures (as opposed to linguistic processing). But that doesn't mean that I draw a lot of relevant things when trying to figure something out or reading a paper. Occasionally I do it, but my usual way of thinking of mathematics is with eyes closed and I just (try to) imagine the corresponding images, which aren't even images in the painter's sense of the word, because, if asked to draw what I "see", I wouldn't be able to do it. I cannot tell what exactly they are, but certainly they are closer to the idea of the visual perception than to anything else despite the fact that the visual perception itself is totally shut down and it is rather about projecting, than about perceiving. I'm inclined to believe that this kind of "perception" would be impossible without having the regular vision machinery in the brain, but again, one needs to run a series of experiments to confirm or to reject this belief.
Also even vision and hearing as such are mainly used for communication rather than for doing mathematics per se. Their only role in mathematical processing is that of a cable connecting the "external hard drive" (usually a piece of paper or a blackboard) with the brain CPU, which doesn't seem to have much memory of its own, just 5-10 registers (my brain definitely works using at most 5: when looking at a non-structured algebraic expression with 6 variables, I cannot remember and reproduce it without compression (a.k.a. finding patterns). This role is very important but, evidently, it can be delegated somewhere else and one can learn to use the internal hard drive too, though I have never managed to do it for random information, just for storage, cataloging, and access to something elusive I call "general ideas", which, by the way, are (next to) impossible to store and transmit using vision or hearing in the form they are really stored (IMHO, those are big "association tables" though, perhaps, "labeled and colored hypergraphs" would be a better choice of words).
I do not roll on the floor or punch walls to do mathematics, but I can easily do either one to vent off my frustration about my ultimate stupidity and total inability to figure the most trivial things out. Those actions are purely emotional.
Finally, I'm as much of a Platonist as far as mathematics is concerned as one can be. So, I believe that all those mathematical objects and constructs are more fundamental or "real" than the so called "real world". When working on a problem, you are just navigating that parallel universe. The words like "I have made half-way to the solution" and "I cannot find the path from A to B" are not metaphors for me. But if you ask me to tell what exactly are the means of landscape recognition, navigation, and propulsion in that parallel Universe, the only thing I'll be able to do (which I have done a few times in front of my students) will be to say "OK, throw a problem at me and I'll show you my thinking". Gowers tried to make a video like that (about computing some determinant), but after first 5 minutes that was a disaster: the first two steps essentially lead to a full solution and then something strange was happening for the next half an hour or so. I still cannot understand what went wrong with him at that moment. My experiments were occasionally more interesting but I doubt they conveyed much: we've all seen a snake effortlessly slithering on the grass. We can easily find the descriptions of the corresponding mechanics and read them. Now lie on the floor and repeat the feat. The same with the senses used. Suppose Terry Tao tells you what he uses exactly and you see it on a brain scan. Will you be ready to prove even some of his already published theorems, not mentioning the ones he hasn't done yet? I won't.
So my final answer is "Who cares?". Use whatever you can and want, play it with no hands barred, feel free to experiment in both work and communication anywhere within the boundaries of common sense and slightly beyond (I once showed a cartoon to my class where the left half was the map with a moving observer and his field of vision and the right half was what he saw, when we studied "linear algebra with application" and I wanted to talk a bit about perspective and computer animation) and just see what works and what doesn't when you do it. As to "what mathematics is, really?" (it looks like we'll slide at least a bit in that direction in this or any other similar soft discussion), as far as I'm concerned, believe in anything you want to believe from Platonism to the Social Construct theory, but remember that the only way you can convert other intelligent people into your faith is by showing that you are a better craftsman than they, and with some of the most interesting characters holding the most outlandish beliefs it may turn out somewhat difficult :-)