  We obtain four Hecke-type double sums for three of Ramanujan's third order
mock theta functions. We discuss how these four are related to the new mock
theta functions of Andrews' work on $q$-orthogonal polynomials and Bringmann,
Hikami, and Lovejoy's work on unified Witten-Reshetikhin-Turaev invariants of
certain Seifert manifolds. We then prove identities between these new mock
theta functions by first expressing them in terms of the universal mock theta
function.
