  In nonlinear greedy approximation theory, bidemocratic bases have
traditionally played the role of dualizing democratic, greedy, quasi-greedy, or
almost greedy bases. In this article we shift the viewpoint and study them for
their own sake, just as we would with any other kind of greedy-type bases. In
particular we show that bidemocratic bases need not be quasi-greedy, despite
the fact that they retain a strong unconditionality flavor which brings them
very close to being quasi-greedy. Our constructive approach gives that for each
$1<p<\infty$ the space $\ell_p$ has a bidemocratic basis which is not
quasi-greedy. We also present a novel method for constructing conditional
quasi-greedy bases which are bidemocratic, and provide a characterization of
bidemocratic bases in terms of the new concepts of truncation quasi-greediness
and partially democratic bases.
