  We present a new description of the spectrum of the (spin-) Dirac operator
$D$ on lens spaces. Viewing a spin lens space $L$ as a locally symmetric space
$\Gamma\backslash \operatorname{Spin}(2m)/\operatorname{Spin}(2m-1)$ and
exploiting the representation theory of the $\operatorname{Spin}$ groups, we
obtain explicit formulas for the multiplicities of the eigenvalues of $D$ in
terms of finitely many integer operations. As a consequence, we present
conditions for lens spaces to be Dirac isospectral. Tackling classic questions
of spectral geometry, we prove with the tools developed that neither spin
structures nor isometry classes of lens spaces are spectrally determined by
giving infinite families of Dirac isospectral lens spaces. These results are
complemented by examples found with the help of a computer.
