Systems for shoulder prostheses

A modular reverse shoulder prosthesis according to embodiments of the present invention includes a stem having a proximal taper and a primary stem axis, the proximal taper extending from the stem about a metaphyseal axis, the metaphyseal axis at an angle with respect to the primary stem axis, a metaphysis having a proximal end, a distal end, a first aperture in the distal end configured to be placed over the proximal taper, and a second aperture in the proximal end having an insert axis that is eccentrically offset from the metaphyseal axis, the metaphysis configured for attachment to the stem at any rotational position of the metaphysis about the metaphyseal axis, and a reverse insert, the reverse insert having a proximal end and a distal end, wherein the proximal end comprises a concave cup formed about a cup axis and configured to receive a glenosphere, and wherein the distal end comprises a locking protrusion, wherein the locking protrusion has an outer surface with a cross-sectional shape that is rotationally symmetrical about the insert axis with respect to a corresponding inner surface of the second aperture, wherein the rotational symmetry has an order of six, seven, eight, nine, or ten.