We study the symmetric rendezvous search problem in which two robots that are unaware of each other's locations try to meet as quickly as possible. In the symmetric version of this problem, the robots are required to execute the same strategy. First, we present a symmetric rendezvous strategy for the robots that are initially placed on the open plane and analyze its competitive performance. We show that the competitive complexity of our strategy is O(d/R) where d is the initial distance between the robots and R is the communication radius. Second, we extend the symmetric rendezvous strategy for the open plane to unknown environments with polygonal obstacles. The extended strategy guarantees a complete coverage of the environment. We analyze the strategy for square, translating robots and show that the competitive ratio of the extended strategy is O(d/D) where D is the length of the sides of the robots. In obtaining this result, we also obtain an upper bound on covering arbitrary polygonal environments which may be of independent interest.