Stephen Smale famously proved in [Trans. Amer. Math. Soc. 90 (1959), 281-290] that any two $C^2$ immersions $S^2\to\mathbb R^3$ are regularly homotopic. This is how we knew that one can do a sphere eversion before ever constructing such a thing, for example.

Later, Morris Hirsch wrote [Trans. Amer. Math. Soc. 93 (1959), 242-276], where he moves from the problem of immersing spheres to immersing general manifolds. His paper is based on Smale's, he says, in roughly the same way that obstruction theory is based on the theory of homotopy groups.

In his paper Hirsch uses his results to solve various problems in immersion theory, but I cannot find in it, nor after a google search, an answer to this:

What is the classification up to regular homotopy of immersions $M\to\mathbb{R^3}$ from a closed oriented surface?

In particular, is there just a single regular homotopy class of immersions of a closed orientable surface of positive genus? Can one everse a sphere with handles?

**Later.** Looking at the explicit construction of representatives of regular homotopy classes of immersions given in the paper by Joel Hass and John Hughes that Danny Ruberman mentions in his answer, the following question asks itself. In some sense, it is closer to the eversion problem, as when turning spheres inside-out we start and end with embeddings.

What is the classification up to regular homotopy of embeddings $M\to\mathbb{R^3}$ of a closed oriented surface?

Of course, along the homotopy the embedding may degrade into just an immersion. Corollary 3.3 in the paper states that if $M$ has genus $g$, then of the $4^g$ regular homotopy classes of immersions $M\to M\times[0,1]$ which are homotopic to the the obvious map $\iota:p\in M\mapsto (p,0)\in M\times [0,1]$ only one can be represented by an embedding, the one which contains $\iota$. So the second question above is what happens when we replace $M\times[0,1]$ by $\mathbb{R^3}$.