This research reports a search for peculiar monobridged structures of the E2H2 molecules (E = Be, Mg, Ca, Sr, Ba). For Be2H2 and Mg2H2, the monobridged geometry is not an equilibrium but rather a transition state between the vinylidene-like structure and the global minimum HE-EH linear geometry. However, for Ca2H2, Sr2H2, and Ba2H2, this situation changes significantly; the linear structure is no longer the global minimum but lies higher in energy than two other equilibria, the dibridged and monobridged structures. The planar dibridged structures of both Sr2H2 and Ba2H2 should be observable via IR spectroscopy. Although the remarkable monobridged structures lie 8.3 (Sr) and 7.6 kcal/mol (Ba) higher, the large IR intensities of the terminal E-H stretching frequencies may make the monobridged structures observable. The monobridged structures have sizable permanent dipole moments (3.07 and 3.06 D for Sr and Ba, respectively) and also should be observable via microwave spectroscopy.