  In our previous works, we have proposed a quantum description of relativistic
orientable objects by a scalar field on the Poincar\'{e} group. This
description is, in a sense, a generalization of ideas used by Wigner, Casimir
and Eckart back in the 1930's in constructing a non-relativistic theory of a
rigid rotator. The present work is a continuation and development of the above
mentioned our works. The position of the relativistic orientable object in
Minkowski space is completely determined by the position of a body-fixed
reference frame with respect to the space-fixed reference frame, and can be
specified by elements $q$ of the motion group of the Minkowski space - the
Poincar\'e group $M(3,1)$. Quantum states of relativistic orientable objects
are described by scalar wave functions $f(q)$ where the arguments $q=(x,z)$
consist of Minkowski space-time points $x$, and of orientation variables $z$
given by elements of the matrix $Z\in SL(2,C)$. Technically, we introduce and
study the so-called double-sided representation
$\boldsymbol{T}(\boldsymbol{g})f(q)=f(g_l^{-1}qg_r)$,
$\boldsymbol{g}=(g_l,g_r)\in \boldsymbol{M}$, of the group $\boldsymbol{M}$, in
the space of the scalar functions $f(q)$. Here the left multiplication by
$g_l^{-1}$ corresponds to a change of space-fixed reference frame, whereas the
right multiplication by $g_r$ corresponds to a change of body-fixed reference
frame. On this basis, we develop a classification of the orientable objects and
draw the attention to a possibility of connecting these results with the
particle phenomenology. In particular, we demonstrate how one may identify
fields described by linear and quadratic functions of $z$ with known elementary
particles of spins $0$,$\frac{1}{2}$, and $1$. The developed classification
does not contradict the phenomenology of elementary particles and, moreover, in
some cases give its group-theoretic explanation.
