  In a previous paper, we considered the motion of massive spinning test
particles in the "pole-dipole" approximation, as described by the
Mathisson--Papapetrou--Dixon (MPD) equations, and examined its properties in
dependence on the spin supplementary condition. We decomposed the equations in
the orthonormal tetrad based on the time-like vector fixing the spin condition
and on the corresponding spin, while representing the curvature in terms of the
Weyl scalars obtained in the Newman--Penrose (NP) null tetrad naturally
associated with the orthonormal one; the projections thus obtained did not
contain the Weyl scalars $\Psi_0$ and $\Psi_4$. In the present paper, we choose
the interpretation tetrad in a different way, attaching it to the tangent
$u^\mu$ of the world-line representing the history of the spinning body.
Actually {\em two} tetrads are suggested, both given "intrinsically" by the
problem and each of them incompatible with one specific spin condition. The
decomposition of the MPD equation, again supplemented by writing its right-hand
side in terms of the Weyl scalars, is slightly less efficient than in the
massive case, because $u^\mu$ cannot be freely chosen (in contrast to $V^\mu$)
and so the $u^\mu$-based tetrad is less flexible. In the second part of this
paper, a similar analysis is performed for massless spinning particles; in
particular, a certain "intrinsic" interpretation tetrad is again found. The
respective decomposition of the MPD equation of motion is considerably simpler
than in the massive case, containing only $\Psi_1$ and $\Psi_2$ scalars and not
the cosmological constant. An option to span the spin-bivector eigen-plane,
besides the world-line null tangent, by a main principal null direction of the
Weyl tensor can lead to an even simpler result.
