When can you reverse the orientation of a complex manifold and still get a complex manifold?

If you take an odd dimensional complex manifold $X$ with holomorphic structure $J$ then $-J$ defines on $X$ a holomorphic structure as well. And, of course, $J$ and $-J$ induce on $X$ opposite orientations. In general it is not true that these two complex manifolds are biholomorphic. Indeed, if $X$ is a complex curve, then $(X,J)$ is biholomorphic to $(X,-J)$ only if $X$ admits an anti-holomorphic involution (this will be the case for example if $X$ is given by an equation with real coefficients).

Starting from this example on can construct a (singular) affine variety $Y$ of dimension $3$, such that $(Y,J)$ is not biholomorphic to $(Y,-J)$. Namely, let $C$ be a compact complex curve that does not admit an anti-holomorphic involution say of genus $g=2$. Consider the rank two bundle over it, equal to the sum $TC\oplus TC$ ($TC$ is the tangent bundle to $C$). Contract the zero section of the total space of this bundle, this gives you desired singular $Y$.

It seems to me that you could be interested in the following (I haven't checked the paper in detail, but I think theorems of this "style" could be helpful for you):

• Dieter Kotschick, Orientations and geometrisations of compact complex surfaces (Bull. London Math. Soc. 29 (1997), no. 2, 145–149.)

Theorem Let $X$ be a compact complex surface admitting a complex structure for $\bar{X}$. Then $X$ (and $\bar{X}$) satisfies one of the following:
(1) $X$ is geometrically ruled, or
(2) the Chern numbers $c_1^2$ and $c_2$ of $X$ vanish, or
(3) $X$ is uniformised by the polydisk.
In particular, the signature of $X$ vanishes.

Other material that could be helpful is:

• Dieter Kotschick, Orientation-reversing homeomorphisms in surface geography (Math. Ann. 292 (1992), no. 2, 375–381.)
• Arnaud Beauville, Surfaces complexes et orientation (Astérisque 126 (1985), 41–43.)