  In this paper, we show that quantum twist maps, introduced by
Lenagan-Yakimov, induce bijections between dual canonical bases of quantum
nilpotent subalgebras. As a corollary, we show the unitriangular property
between dual canonical bases and Poincar\'e-Birkhoff-Witt type bases under the
"reverse" lexicographic order. We also show that quantum twist maps induce
bijections between certain unipotent quantum minors.
