I know that the Conway polynomial of a trefoil knot with all negative crossings is 1 + x^2.
I was therefore wondering, would the Conway polynomial of an equivalent trefoil knot with all positive crossings also be 1 + x^2?
Is there a general theorem for converting between the conway polynomials of equivalent knots with all positive or all negative crossings?
For a reference, you might consider looking at chapter 8 of Lickorish's book An Introduction to Knot Theory. In short, the Conway polynomial is from a normalized Alexander polynomial, and the Alexander polynomial is invariant under mirror image. I think it's Fox's A Quick Trip Through Knot Theory that gives a nice reason for this, but this follows from the fact that the Wirtinger presentation can have relations that are either from loops above or below a crossing, and one can pass between these group presentations by formally replacing each generator with its inverse; since these are group presentations for the same group, this is reflected in the Alexander polynomial by a $t\leftrightarrow t^{-1}$ symmetry.
A definition of the Conway polynomial is that it is $\det(t^{1/2}A-t^{-1/2}A^T)$ with $x=t^{-1/2}-t^{1/2}$, where $A$ is a Seifert matrix for a given link, which for a Seifert surface $S\subset S^3$ is the matrix of a form $H_1(S)\times H_1(S)\to \mathbb{Z}$ given by $([\alpha],[\beta])\mapsto \operatorname{lk}(\alpha,\beta^+)$ when $\alpha,\beta$ are simple closed curves and $\beta^+$ is the curve pushed off $S$ in the positive normal direction with respect to the orientation of the surface. The corresponding Seifert matrix for the mirror image of everything is $-A^T$, as outlined in this answer: Seifert Matrix of Amphichiral Knots.
Thus, the Conway polynomial of a mirror image is \begin{align} \det(t^{1/2}(-A^T)-t^{-1/2}(-A^T)^T)&=\det(-(t^{1/2}A-t^{-1/2}A^T)^T)\\ &=(-1)^n\det(t^{1/2}A-t^{-1/2}A^T) \end{align} where $n=\operatorname{rank}(H_1(S))$, which at least for a knot is even. Hence, for a knot the Conway polynomial is invariant under mirror images. (In general, this is true for links with an odd number of components.)
So: the Conway polynomial of a knot and its mirror image are the same. In particular, the two chiral forms of the trefoil both have $1+x^2$ as their Conway polynomials.