  The two-fold singularity has played a significant role in our understanding
of uniqueness and stability in piecewise smooth dynamical systems. When a
vector field is discontinuous at some hypersurface, it can become tangent to
that surface from one side or the other, and tangency from both sides creates a
two-fold singularity. The local flow bears a superficial resemblance to
so-called {\it folded singularities} in (smooth) slow-fast systems, which arise
at the intersection of attractive and repelling branches of slow invariant
manifolds, important in the local study of canards and mixed mode oscillations.
Here we show that these two singularities are intimately related. When the
discontinuity in a piecewise smooth system is regularized (smoothed out) at a
two-fold singularity, the resulting system can be mapped onto a folded
singularity. The result is not obvious since it requires the presence of
nonlinear or `hidden' terms at the discontinuity, which turn out to be
necessary for structural stability of the regularization (or smoothing) of the
discontinuity, and necessary for mapping to the folded singularity.
