We study low-frequency waves and instabilities in collisionless plasmas characterized by a gyrotropic (P⊥ ≠ P∥) thermal pressure and embedded in a magnetized, stratified, and rotating environment. We consider only small-scale perturbations (with a wavelength that is short compared with the stratification scale height), and we make the restrictive assumption that the equilibrium parameters are invariant along magnetic field lines. In addition, we focus on two-component ion-electron plasmas, deferring the treatment of multicomponent plasmas to a companion paper. Our approach is not pure magnetohydrodynamics (MHD) but instead a mixture of MHD and kinetics, insofar as all our equations are fluid except the equations of state for P⊥ and P∥, whose perturbed expression is deduced from the Vlasov equation. The emphasis of the paper is laid on the so-called quasi-interchange modes, i.e., the modes on which stratification has a significant impact. After showing that these modes consist of the Alfv¿n, slow, and mirror modes at nearly perpendicular wave vectors, we investigate the question of their stability in a nonrotating system.