  We construct an algorithm for computing the cycle classes of the spin
components of a stratum of differentials in the moduli space of stable curves
$\overline{\mathcal{M}}_{g,n}$. In addition, we implement it within the Sage
package admcycles. Our main strategy is to reconstruct these cycles by their
restrictions to the boundary of $\overline{\mathcal{M}}_{g,n}$ via clutching
maps. These restrictions can be computed recursively by smaller dimensional
spin classes and determine the original class via a certain system of linear
equations. To study the spin parities on the boundary of a stratum of
differentials of even type, we make use of the moduli space of multi-scale
differentials introduced in [BCCGM19].
  As an application of our algorithm, one can verify a conjecture on spin
double ramification cycles stated in [CSS21] in many examples, by using the
results computed by our algorithm.
