Primitive word and cyclic rotations
Yes, this is true. The direct implication ($\Rightarrow$) is the hardest to prove, and you will need the following lemma: two words commute if and only if they are powers of the same word (R.C. Lyndon and M.P. Schützenberger, The equation a M = bncp in a free group, Michigan Math. J. 
9 (1962) 289-298.  Or in this: H. Petersen, On the language of primitive words, Theor. Comp. Sci., 1996 https://doi.org/10.1016/0304-3975(95)00098-4). The rest of the proof should be straightforward enough.