This paper investigates the generalized multi-period mean-variance investment-reinsurance optimization model in a discrete-time framework for a general insurance company that contains a reinsurer and an insurer. The intertemporal restrictions and the common interests of the reinsurer and the insurer are considered. The common goal of the reinsurer and the insurer is to maximize the expectation of the weighted sum of their wealth processes and minimize the corresponding variance. Based on the game method, we obtain the Nash equilibrium investment-reinsurance strategies for the above-proposed model and find out the equilibrium strategies when unilateral interest is considered. In addition, the Nash equilibrium investment-reinsurance strategies are deduced under two special premium calculated principles (i.e., the expected value premium principle and the variance value premium principle). We theoretically study the effect of the intertemporal restrictions on Nash equilibrium investment-reinsurance strategies and find the effect depends on the value of some parameters, which differs from the previous researches that generally believed that intertemporal restrictions would make investors avoid risks. Finally, we perform corresponding numerical analyses to verify our theoretical results.