Let $C_{2k_1, 2k_2, \ldots, 2k_t}$ denote the graph obtained by intersecting $t$ distinct even cycles $C_{2k_1}, C_{2k_2}, \ldots, C_{2k_t}$ at a unique vertex. In this paper, we determine the unique graphs with maximum adjacency spectral radius among all graphs on $n$ vertices that do not contain any $C_{2k_1, 2k_2, \ldots, 2k_t}$ as a subgraph, for $n$ sufficiently large. When one of the constituent even cycles is a $C_4$, our results improve upper bounds on the Turán numbers for intersecting even cycles that follow from more general results of Füredi [20] and Alon, Krivelevich and Sudakov [1]. Our results may be seen as extensions of previous results for spectral Turán problems on forbidden even cycles $C_{2k}, k\ge 2$ (see [8, 34, 44, 45]).