Immersions of surfaces in $\mathbb{R}^3$

(Based on the comment of Mariano Suárez-Álvarez, there was a false assumption in my original answer. This is an attempt to correct it.)

1) Let $M$ be a closed smooth manifold with $k < n$. According to Smale-Hirsch theory, the space of immersions $M^k \to \Bbb R^n$ is homotopy equivalent to the space of tangent bundle monomorphisms $TM\to \Bbb R^n$, by which I mean the space of sections of the fiber bundle over $M$ whose fiber at $x\in M$ is given by the space of linear injections $T_xM\to \Bbb R^n$. If $M$ is oriented, then this last space is identified with the space of linear orientation preserving isomorphisms $T_xM \oplus \Bbb R \to \Bbb R^n$.

2) In (1) there is no loss in assuming $M$ is equipped with a Riemannian metric and that the isomorphisms $T_xM \oplus \Bbb R \to \Bbb R^n$ are linear isometries.

3) Assume $k=2$ and $M$ is oriented. Then the space of immersions $I(M,\Bbb R^3)$ is identified with the space of sections of fiber bundle over $E\to M$ whose fiber at $x\in M$ is the space of orientation preserving linear isometries $T_x M\oplus \Bbb R \to \Bbb R^3$.

4) Any oriented surface $M$ is stably parallelizable since it embeds in $\Bbb R^3$. So a choice of stable parallelization gives a preferred isomorphism $T_x M \oplus \Bbb R\cong\Bbb R^3$. So the bundle in $(2)$ is trivializable: $E \cong M \times SO(3)$. Hence the space of sections coincides with the space of maps $\text{map}(M,SO(3))$.

Hence there is a bijection between the homotopy classes of immersions of $M$ in $\Bbb R^3$ and the set of homotopy classes $[M,SO(3)]$.

5) $SO(3) \cong \Bbb RP^3$ and the map $\Bbb RP^3 \to \Bbb RP^\infty$ is $3$-connected. So $[M,SO(3)] \cong H^1(M,\Bbb Z_2) \cong \oplus_{2g} \Bbb Z_2$, where $g$ is the genus.

A paper of Joel Hass and John Hughes (Immersions of surfaces in 3-manifolds. Topology 24 (1985), no. 1, 97–112) gives a similar calculation where the target is an arbitrary 3-manifold. A bonus, which is of interest even in the case when the target is $\mathbb{R}^3$, is an explicit geometric representative for each regular homotopy class of immersions (in a given homotopy class).

Regarding the classification up to regular homotopy of embeddings, U. Pinkall proves in [Topology 24 (1984), 421–434] that

if $f$, $g:M\to\mathbb R^3$ are two embeddings of a compact orientable surface, then there is a diffeomorphism $h:M\to M$ such that $f$ and $g\circ h$ are regularly homotopic.

This is not quite as good as «one can everse a compact orientable surface», though: that the usual embedding $f:S^2\to\mathbb R^3$ and $g=-f$ are related as in Pinkall's theorem is obvious.

Pinkall notes that it is not true that two embeddings $f$, $g:M\to\mathbb R^3$ are necesarily regularly homotopic, but does not say anything how often they are.