  We determine the critical exponent and the recurrence function of
complementary symmetric Rote sequences. The formulae are expressed in terms of
the continued fraction expansions associated with the S-adic representations of
the corresponding standard Sturmian sequences. The results are based on a
thorough study of return words to bispecial factors of Sturmian sequences.
Using the formula for the critical exponent, we describe all complementary
symmetric Rote sequences with the critical exponent less than or equal to 3,
and we show that there are uncountably many complementary symmetric Rote
sequences with the critical exponent less than the critical exponent of the
Fibonacci sequence. Our study is motivated by a~conjecture on sequences rich in
palindromes formulated by Baranwal and Shallit. Its recent solution by Curie,
Mol, and Rampersad uses two particular complementary symmetric Rote sequences.
