The epitaxial relationship in the thermal phase transition between double gyroid (DG) and hexagonally packed cylinder (HEX) phases in polystyrene-block-polyisoprene thin films on Si wafer was investigated using transmission electron microtomography and grazing incidence small-angle X-ray scattering. Two different types of epitaxial transitions were observed, and they appeared to be selectively favored depending on the transition direction. One type of epitaxial relationship prevails in the phase transition from DG to HEX upon heating in which {121}(DG), {111}(DG), and {220}(DG) are converted to {100}(HEX), {110}(HEX), and {001}(HEX), respectively. The interphase planes are {220}(DG) and {001}(HEX), and the cylinders meet the {220}(DG) plane perpendicularly (head-on, Type A) at the grain boundary between DG and HEX. Although there are small dimensional mismatch and distortion in the location of the cylinders in this epitaxial relationship, all cylinders are formed along the topologically equivalent DG skeletal path. On the other hand, in the transition from HEX to DG upon cooling, another epitaxial relationship as well as the head-on type epitaxy was observed, in which {100}(HEX), {110}(HEX), and {001}(HEX) are converted to {121}(DG), {220}(DG), and {111}(DG), respectively. The interphase planes are {220}(DG) and {110}(HEX), and the cylinders meet the {220}DG plane in parallel (side-on, Type B) at the grain boundary between HEX and DG. The domain spacing and the symmetry of the two phases are matched near perfectly, but cylinders are converted to two different DG skeletal paths. The Type B epitaxy is hardly observed in the transition from DG to HEX.