  We study twistor spinors (with torsion) on Riemannian spin manifolds $(M^{n},
g, T)$ carrying metric connections with totally skew-symmetric torsion. We
consider the characteristic connection $\nabla^{c}=\nabla^{g}+\frac{1}{2}T$ and
under the condition $\nabla^{c}T=0$, we show that the twistor equation with
torsion w.r.t. the family $\nabla^{s}=\nabla^{g}+2sT$ can be viewed as a
parallelism condition under a suitable connection on the bundle
$\Sigma\oplus\Sigma$, where $\Sigma$ is the associated spinor bundle.
Consequently, we prove that a twistor spinor with torsion has isolated zero
points. Next we study a special class of twistor spinors with torsion, namely
these which are $T$-eigenspinors and parallel under the characteristic
connection; we show that the existence of such a spinor for some $s\neq 1/4$
implies that $(M^{n}, g, T)$ is both Einstein and $\nabla^{c}$-Einstein, in
particular the equation ${\rm Ric}^{s}=\frac{{\rm Scal}^{s}}{n}g$ holds for any
$s\in\mathbb{R}$. In fact, for $\nabla^{c}$-parallel spinors we provide a
correspondence between the Killing spinor equation with torsion and the
Riemannian Killing spinor equation. This allows us to describe 1-parameter
families of non-trivial Killing spinors with torsion on nearly K\"ahler
manifolds and nearly parallel ${\rm G}_2$-manifolds, in dimensions 6 and 7,
respectively, but also on the 3-dimensional sphere ${\rm S}^{3}$. We finally
present applications related to the universal and twistorial eigenvalue
estimate of the square of the cubic Dirac operator.
