  We extend the Gordon-Litherland pairing to links in thickened surfaces, and
use it to define signature, determinant, and nullity invariants for links that
bound (unoriented) spanning surfaces. The invariants are seen to depend only on
the $S^*$-equivalence class of the spanning surface. We prove a duality result
relating the invariants from one $S^*$-equivalence class of spanning surfaces
to the restricted invariants of the other.
  Using Kuperberg's theorem, these invariants give rise to well-defined
invariants of checkerboard colorable virtual links. The determinants can be
applied to determine the minimal support genus of a checkerboard colorable
virtual link. The duality result leads to a simple algorithm for computing the
invariants from the Tait graph associated to a checkerboard coloring. We show
these invariants simultaneously generalize the combinatorial invariants defined
by Im, Lee, and Lee, and those defined by Boden, Chrisman, and Gaudreau for
almost classical links.
  We examine the behavior of the invariants under orientation reversal, mirror
symmetry, and crossing change. We give a 4-dimensional interpretation of the
Gordon-Litherland pairing by relating it to the intersection form on the
relative homology of certain double branched covers. This correspondence is
made explicit through the use of virtual linking matrices associated to
(virtual) spanning surfaces and their associated (virtual) Kirby diagrams.
