  We introduce and study deformation $T_{{\bf b},\phi}$ of Minkowski norms in
$\mathbb{R}^n$, determined by a set ${\bf b}=(\beta_1,\ldots,\beta_p)$ of
linearly independent 1-forms and a smooth positive function $\phi$ of $p$
variables. In particular, the $T_{{\bf b},\phi}$-image of a Euclidean norm
$\alpha$ is a Minkowski norm, whose indicatrix is a rotation hypersurface with
a $p$-dimensional axis passing through the origin. For $p=1$, our deformation
generalizes construction of $(\alpha,\beta)$-norm; the last ones form a rich
class of "computable" Minkowski norms and play an important role in Finsler
geometry. We use compositions of $T_{{\bf b},\phi}$-deformations with ${\bf
b}$'s of length $p$ to define an equivalence relation $\overset{p}\sim$ on the
set of all Minkowski norms in $\mathbb{R}^n$. We apply M. Matsumoto result to
characterize the cases when the Cartan torsions of a norm and its $T_{{\bf
b},\phi}$-image either coincide or differ by a $C$-reducible term.
