Return words in codings of rotations

We study the structure of infinite words obtained by coding rotations on partitions of the unit circle by inspecting the return words. The main result is that every factor of a coding of rotations on two intervals has at most 4 complete return words, where the bound is realized only for a finite number of factors. As a byproduct we obtain that when the partition consists of two intervals, then the corresponding word is full, that is, it realizes the maximal palindromic complexity. We also provide a combinatorial proof for the special case of complementary-symmetric Rote sequences by considering both the palindromes and the antipalindromes occurring in it