A piston construction for an opposed-piston engine accommodates contoured end surfaces of the piston crowns and articulation of the pistons with piston rods. The shape of each piston crown includes a bowl with an adjoining ridge that protrudes axially of the piston. A cooling construction for the piston includes an outer gallery running around the inner surface of the piston sidewall in the vicinity of the crown. The outer gallery girds and is in fluid communication with a central gallery that abuts the deepest part of the bowl. The outer gallery has an asymmetric profile that rises under the ridge and that slants upwardly under a peripheral portion of the bowl. A mechanism for coupling the piston to a piston rod includes a biaxial slipper bearing.