  Experimental results show that, when the order $n$ is odd, there are de
Bruijn sequences such that the corresponding complement sequence and the
reverse sequence are the same. In this paper, we propose one efficient method
to generate such de Bruijn sequences. This solves an open problem asked by
Fredricksen forty years ago for showing the existence of such de Bruijn
sequences when the odd order $n >1$. Moreover, we refine a characterization of
de Bruijn sequences with the same complement and reverse sequences and study
the number of these de Bruijn sequences, as well as the distribution of de
Bruijn sequences of the maximum linear complexity.
