Ca2Co2O5 in the brownmillerite form was synthesized using a high-pressure optical-image floating zone furnace, and single crystals with dimensions up to 1.4x0.8x0.5 mm3 were obtained. At room temperature, Ca2Co2O5 crystallizes as a fully ordered brownmillerite variant in the orthorhombic space group Pcmb (No. 57) with unit cell parameters a=5.28960(10) {\AA}, b=14.9240(2) {\AA}, and c=10.9547(2) {\AA}. With decreasing temperature, it undergoes a re-entrant sequence of first-order structural phase transitions (Pcmb to P2/c11 to P121/m1 to Pcmb) that is unprecedented among brownmillerites, broadening the family of space groups available to these materials and challenging current approaches for sorting the myriad variants of brownmillerite structures. Magnetic susceptibility data indicate antiferromagnetic ordering in Ca2Co2O5 occurs near 240 K, corroborated by neutron powder diffraction. Below 140 K, the specimen exhibits a weak ferromagnetic component directed primarily along the b axis that shows a pronounced thermal and magnetic history dependence.