We propose a theory for the underdoped hole-doped cuprates, focusing on the "nodal-antinodal dichotomy" observed in recent experiments. Our theory begins with an ordered antiferromagnetic Fermi liquid with electron and hole pockets. We argue that it is useful to consider a quantum transition at which the loss of antiferromagnetic order leads to a hypothetical metallic "algebraic charge liquid" (ACL) with pockets of charge -e and +e fermions, and an emergent U(1) gauge field; the instabilities of the ACL lead to the low-temperature phases of the underdoped cuprates. The pairing instability leads to a superconductor with the strongest pairing within the -e Fermi pockets, a d-wave pairing signature for electrons, and very weak nodal-point pairing of the +e fermions near the Brillouin-zone diagonals. The influence of an applied magnetic field is discussed using a proposed phase diagram as a function of field strength and doping. We describe the influence of gauge field and pairing fluctuations on the quantum Shubnikov-de Haas oscillations in the normal states induced by the field. For the finite-temperature pseudogap region, our theory has some similarities to the phenomenological two-fluid model of -2e bosons and +e fermions proposed by Geshkenbein [Phys. Rev. B 55, 3173 (1997)], which describes anomalous aspects of transverse transport in a magnetic field.