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: Part of [Kotschick97, Thm. 2]
Let X be a compact complex surface admitting a complex structure for −X. Then 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.)