  We discuss a discrete analogue of the Dirac-K\"{a}hler equation in which
chiral properties of the continual counterpart are captured. We pay special
attention to a discrete Hodge star operator. To build one a combinatorial
construction of double complex is used. We describe discrete exterior calculus
operations on a double comlex and obtain the discrete Dirac-K\"{a}hler equation
using these tools. Self-dual and anti-self-dual discrete inhomogeneous forms
are presented. The chiral invariance of the massless discrete Dirac-K\"{a}hler
equation is shown. Moreover, in the massive case we prove that a discrete
Dirac-K\"{a}hler operator flips the chirality.
