  Marked vertex diagrams provide a combinatorial way to represent knotted
surfaces in $\mathbb{R}^4$; including virtual crossings allows for a theory of
virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants
are defined only for marked vertex diagrams representing knotted orientable
surfaces; we extend these invariants to all virtual marked vertex diagrams by
considering colorings by involutory biquandles, also known as bikei. We
introduce a way of representing marked vertex diagrams with Gauss diagrams and
use these to characterize orientability.
