Heteroclinic bifurcations with orbit-flips and inclination-flips are investigated in a four-dimensional reversible system by using the method originally established in [Zhu, 1998; Zhu & Xia, 1998]. The existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic orbit, R-symmetric homoclinic orbit and R-symmetric periodic orbit are obtained. The double R-symmetric homoclinic bifurcation is found, and the continuum of R-symmetric periodic orbits accumulating into a homoclinic orbit is also demonstrated. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation diagrams are drawn.