  We consider area minimizing $m$-dimensional currents $\mathrm{mod}(p)$ in
complete $C^2$ Riemannian manifolds $\Sigma$ of dimension $m+1$. For odd moduli
we prove that, away from a closed rectifiable set of codimension $2$, the
current in question is, locally, the union of finitely many smooth minimal
hypersurfaces coming together at a common $C^{1,\alpha}$ boundary of dimension
$m-1$, and the result is optimal. For even $p$ such structure holds in a
neighborhood of any point where at least one tangent cone has
$(m-1)$-dimensional spine. These structural results are indeed the byproduct of
a theorem that proves (for any modulus) uniqueness and decay towards such
tangent cones. The underlying strategy of the proof is inspired by the
techniques developed by Leon Simon in "Cylindrical tangent cones and the
singular set of minimal submanifolds" (J. Diff. Geom. 1993) in a class of
multiplicity one stationary varifolds. The major difficulty in our setting is
produced by the fact that the cones and surfaces under investigation have
arbitrary multiplicities ranging from $1$ to $\lfloor \frac{p}{2}\rfloor$.
