In an agent-based computer model, we simulate the formation and recall of anti-virus immunological memory. Specifically we try to predict what will happen, both to the response and to memory, when the second infecting virus is partly different from the first one, and when the cross-reactivity of the two branches of the immune system (IS), humoral and cellular, is asymmetrical, or "split". The simulations explore systematically epitope distances, and measure all changes in affinity, cellularity and efficiency in clearing the infection. Besides obvious cooperation, they reveal powerful competitions between the branches, and more intriguing, between cross-reacting and new responses when the latter suffer the competition by preformed cell-rich but inefficient clones, as memory, usually an asset, becomes a liability.