  The classical spinning particles are considered such that quantization of
classical model leads to an irreducible massive representation of the
Poincar\'e group. The class of gauge equivalent classical particle world lines
is shown to form a $[(d+1)/2]$-dimensional world sheet in $d$-dimensional
Minkowski space, irrespectively to any specifics of classical model. For
massive spinning particles in $d=3,4$, the world sheets are shown to be
cylinders. The radius of cylinder is fixed by representation. In higher
dimensions, particle's world sheet turn out to be a toroidal cylinder
$\mathbb{R}\times \mathbb{T}^D$, $D=[(d-1)/2]$. Proceeding from the fact that
the world lines of irreducible classical spinning particles are cylindrical
curves, while all the lines are gauge equivalent on the same world sheet, we
suggest a method to deduce the classical equations of motion for particles and
also to find their gauge symmetries. In $d=3$ Minkowski space, the spinning
particle path is defined by a single fourth-order differential equation having
two zero-order gauge symmetries. The equation defines particle's path in
Minkowski space, and it does not involve auxiliary variables. A special case is
also considered of cylindric null-curves, which are defined by a different
system of equations. It is shown that the cylindric null-curves also correspond
to irreducible massive spinning particles. For the higher-derivative equation
of motion of the irreducible massive spinning particle, we deduce the
equivalent second-order formulation involving an auxiliary variable. The
second-order formulation agrees with a previously known spinning particle
model.
