What can one say about the relation between the Hodge decompositions of $H^\*(X,C)$ and $H^{*}(X_\sigma,C)$ for a complex algebraic smooth projective variety $X$ and $\sigma$ an automorphism of the field of complex numbers ? I am also interested in conjectural results, e.g. whether Standard Conjectures imply something about the relationship of the Hodge structure of conjugate varieties $X$ and $X_\sigma$.

For example, it appears that the Hodge numbers $h^{ab}=dim_C H^a(X,\Omega_X^b)$ and $h_\sigma ^{ab}=dim_C H^a(X_\sigma,\Omega_{X_\sigma}^b)$ are necessarily the same (being defined via de Rham cohomology); is this right (cf. discussion in the comments)? what other invariants/subspaces etc are the same or known to be possibly different ? Can one define a related Galois (or $Aut(C/Q)$) action on the Hodge structures, something like where an automorphism $\sigma\in Aut(C/Q)$ takes the Hodge structure of $X$ into that of $X_\sigma$'s ?

I know of several references that show that conjugate varieties may be rather different topologically. F.Charles, Conjugate varieties with distinct real cohomology algebras lists most references I know of and well, proves what's in the title. Serre constructed two two non-homotopic conjugated varieties with different fundamental groups; his proof exploits the fact that for a CM elliptic curve, its fundamental group has different $EndE$-module structures for different embeddings. Abelson constructs two non-homotopic conjugated varieties with the same finite fundamental groups.