  We derive necessary and sufficient conditions for local unitary (LU)
operators to leave invariant the set of 1-qubit reduced density matrices of a
multi-qubit state. LU operators with this property are tensor products of {\it
cyclic local} operators, and form a subgroup, the centralizer subgroup of the
set of reduced states, of the Lie group $SU(2)^{\otimes n}$. The dimension of
this subgroup depends on the type of reduced density matrices. It is maximum
when all reduced states are maximally mixed and it is minimum when none of them
is maximally mixed. For any given multi-qubit state, pure or mixed, we compute
the LU operators that fix the corresponding reduced density matrices and
determine the equivalence class of the given state.
