We study low-frequency waves and instabilities in homogeneous, collisionless plasmas whose thermal pressure is gyrotropic, i.e., characterized by two distinct components, P⊥ and P∥, perpendicular and parallel, respectively, to the magnetic field. In order to obtain the complete dispersion relation of low-frequency waves, we need equations of state for P⊥ and P∥. After showing that these equations differ from the widely used double-adiabatic, or CGL, magnetohydrodynamic (MHD) equations, we derive their correct expressions by resorting to the Vlasov equation. We thus obtain a dispersion relation, which is no longer strictly MHD: in addition to the three standard (fast, slow, and Alfv¿n) modes from double-adiabatic MHD theory, our dispersion relation also contains the mirror mode from kinetic theory. We examine the physical characteristics and stability properties of each of these four modes and provide numerical solutions for their frequency and growth/decay rate, both for simple proton--electron plasmas and for multispecies plasmas.