  For the space $\mathcal{S}$ of $C^3$ quintics on the Powell-Sabin 12-split of
a triangle, we determine the simplex splines in $\mathcal{S}$ and the six
symmetric simplex spline bases that reduce to a B-spline basis on each edge,
have a positive partition of unity, a (barycentric) Marsden identity, and
domain points with an intuitive control net. We provide a quasi-interpolant
with approximation order 6 and a Lagrange interpolant at the domain points. The
latter can be used to show that each basis is stable in the $L_\infty$ norm,
which yields an $h^2$ bound for the distance between the B\'ezier ordinates and
the values of the spline at the corresponding domain points. Finally, for one
of these bases we provide $C^0$, $C^1$, and $C^2$ conditions on the control
points of two splines on adjacent macrotriangles, and a conversion to the
Hermite nodal basis.
