  We present an extensive analytical and numerical study of the
antiferromagnetic Heisenberg model on the Cairo pentagonal lattice, the dual of
the Shastry-Sutherland lattice with a close realization in the S=5/2 compound
Bi2Fe4O9. We consider a model with two exchange couplings suggested by the
symmetry of the lattice, and investigate the nature of the ground state as a
function of their ratio x and the spin S. After establishing the classical
phase diagram we switch on quantum mechanics in a gradual way that highlights
the different role of quantum fluctuations on the two inequivalent sites of the
lattice. The most important findings for S=1/2 include: (i) a surprising
interplay between a collinear and a four-sublattice orthogonal phase due to an
underlying order-by-disorder mechanism at small x (related to an emergent J1-J2
effective model with J2 >> J1), and (ii) a non-magnetic and possibly
spin-nematic phase with d-wave symmetry at intermediate x.
