  The problem on mapping between two Lagrangian descriptions (using a commuting
$c$-number spinor $\psi_{\alpha}$ or anticommuting pseudovector $\xi_{\mu}$ and
pseudoscalar $\xi_5$ variables) of the spin degrees of freedom of a color
spinning massive particle interacting with background non-Abelian gauge field,
is considered. A general analysis of the mapping between a pair of {\it
Majorana} spinors $(\psi_{\alpha},\theta_{\alpha})$ ($\theta_{\alpha}$ is some
auxiliary anticommuting spinor) and a real anticommuting tensor aggregate
$(S,V_{\mu},\!\,^{\ast}T_{\mu \nu},A_{\mu},P)$, is presented. A complete system
of bilinear relations between the tensor quantities, is obtained. The analysis
we have given is used for the above problem of the equivalence of two different
ways of describing the spin degrees of freedom of the relativistic particle.
The mapping of the kinetic term $(i\hbar/2)(\bar{\theta}\theta)
(\dot{\bar{\psi}}\psi-\bar{\psi}\dot{\psi})$, the term
$(1/e)(\bar{\theta}\theta)\dot{x}_{\mu}(\bar{\psi}\gamma^{\mu}\psi)$ that
provides a couple of the spinning variable $\psi$ and the particle velocity
$\dot{x}_{\mu}$, and the interaction term
$\hbar(\bar{\theta}\theta)Q^{a}F_{\mu\nu}^a (\bar{\psi}\sigma^{\mu\nu}\psi)$
with an external non-Abelian gauge field, are considered in detail. In the
former case a corresponding system of bilinear identities including both the
tensor variables and their derivatives
$(\dot{S},\dot{V}_{\mu},\!\,^{\ast}\dot{T}_{\mu\nu},\dot{A}_{\mu}, \dot{P})$,
is defined. A detailed analysis of the local bosonic symmetry of the Lagrangian
with the commuting spinor $\psi_{\alpha}$, is carried out. A connection of this
symmetry with the local SUSY transformation of the Lagrangian containing
anticommuting pseudovector and pseudoscalar variables, is considered.
