In this paper the concept of metaperiodic word of a given exponent is introduced, and transformations (reversals) of such words are considered. It is proved that in a free periodic group $B(m,n)$ of any odd exponent $n\geqslant665$ with $m\geqslant66$ generators an infinite independent system of complementary relations can be singled out. It follows that in $B(m,n)$ there exist infinite ascending and descending chains of normal subgroups and also a recursively defined factor group of $B(m,n)$ with an unsolvable identity problem. Bibliography: 4 titles.