  We study a certain theta lift which maps weight $-2k$ to weight $1/2-k$
harmonic weak Maass forms for $k \in \mathbb{Z}, k \geq 0$, and which is
closely related to the classical Shintani lift from weight $2k+2$ to weight
$k+3/2$ cusp forms. We compute the Fourier expansion of the theta lift and show
that it involves twisted traces of CM values and geodesic cycle integrals of
the input function. As an application, we obtain a criterion for the
non-vanishing of the central $L$-value of an integral weight newform $G$ in
terms of the holomorphicity of the theta lift of a certain harmonic weak Maass
form associated to $G$. Moreover, we derive interesting identities between
cycle integrals of different kinds of modular forms.
