In recent work, Hickerson and the author demonstrated that it is useful to think of Appell--Lerch sums as partial theta functions. This notion can be used to relate identities involving partial theta functions with identities involving Appell--Lerch sums. In this sense, Appell--Lerch sums and partial theta functions appear to be dual to each other. This duality theory is not unlike that found by Andrews between various sets of identities of Rogers-Ramanujan type with respect to Baxter's solution to the hard hexagon model of statistical mechanics. As an application we construct bilateral $q$-series with mixed mock modular behaviour.

To be published in Advances in Mathematics