In this paper, we study the doubly conditional reflected backward stochastic differential equations (BSDEs), where constraints are made on the conditional expectation of the first component of the solution with respect to a general subfiltration. With the help of the Skorokhod problem on a time-dependent interval and the Dynkin game in a general framework, we establish the existence and uniqueness result under the Mokobodski condition for the obstacles. The relation between the conditional expectation of the solution and the value function of a certain Dynkin game with partial information is obtained. As a by-product, we obtain a weaker version of the comparison theorem. Finally, we provide an application to the starting and stopping problem in reversible investments under partial information.