  We introduce a new class of bases for quantized universal enveloping algebras
$U_q(\mathfrak g)$ and other doubles attached to semisimple and Kac-Moody Lie
algebras. These bases contain dual canonical bases of upper and lower halves of
$U_q(\mathfrak g)$ and are invariant under many symmetries including all
Lusztig's symmetries if $\mathfrak g$ is semisimple. It also turns out that a
part of a double canonical basis of $U_q(\mathfrak g)$ spans its center.
