New distorted variants of the cubic BaHg11 structure type have been synthesized in Ga flux. Multiple phases of CePd3+xGa8-x, which include an orthorhombic Pmmn structure (x = 3.21(2)), a rhombohedral R (3) over barm structure (x = 3.13(4)), and a cubic Fm (3) over barm superstructure (x = 2.69(6)), form preferentially depending on reaction cooling rate and isolation temperature. Differential thermal analysis and in situ temperature-dependent powder X-ray diffraction patterns show a reversible phase transition at approximately 640 C between the low temperature orthorhombic and rhombohedral structures and the high temperature cubic superstructure. Single crystal X-ray diffraction experiments indicate that the general structure of BaHg11, including the intersecting planes of a kagome-type arrangement of Ce atoms, is only slightly distorted in the low temperature phases. A combination of Kondo, crystal electric field, and magnetic frustration effects may be present, resulting in low temperature anomalies in magnetic susceptibility, electrical resistivity, and heat capacity measurements. In addition to CePd3+xGa8-x, the rare earth analogues REPd3+xGa8-x, RE = La, Nd, Sm, Tm, and Yb, were successfully synthesized and also crystallize in one of the lower symmetry space groups.