We classify the possible shapes of cosmic string cusps and how they transform under Lorentz boosts. A generic cusp can be brought into a form in which the motion of the cusp tip lies in the plane of the cusp. The cusp whose motion is perpendicular to this plane, considered by some authors, is a special case and not the generic situation. We redo the calculation of the energy in the region where the string overlaps itself near a cusp, which is the maximum energy that can be released in radiation. We take into account the motion of a generic cusp and the resulting Lorentz contraction of the string core. The result is that the energy scales as $\sqrt{\mathrm{rL}}$ instead of the usual ${r}^{1/3}{L}^{2/3},$ where r is the string radius and L is the typical length scale of the string. Since $r\ensuremath{\ll}L$ for cosmological strings, the radiation is strongly suppressed and could not be observed.