  The spectral properties of signed directed graphs, which may be naturally
obtained by assigning a sign to each edge of a directed graph, have received
substantially less attention than those of their undirected and/or unsigned
counterparts. To represent such signed directed graphs, we use a striking
equivalence to $\mathbb{T}_6$-gain graphs to formulate a Hermitian adjacency
matrix, whose entries are the unit Eisenstein integers $\exp(k\pi i/3),$ $k\in
\mathbb{Z}_6.$ Many well-known results, such as (gain) switching and eigenvalue
interlacing, naturally carry over to this paradigm. We show that non-empty
signed directed graphs whose spectra occur uniquely, up to isomorphism, do not
exist, but we provide several infinite families whose spectra occur uniquely up
to switching equivalence. Intermediate results include a classification of all
signed digraphs with rank $2,3$, and a deep discussion of signed digraphs with
extremely few (1 or 2) non-negative (eq. non-positive) eigenvalues.
