EDIT: I have now asked the same question on MO.
Background. Let $G=SL(2,\mathbb{R})$, let $K=SO(2)$, and let $\Gamma$ be a lattice in $G$, e.g. $SL(2,\mathbb{Z})$. Let $\phi \in L^1(G)$ be $\mathfrak{Z}$-finite, and let the Poincaré series $P_\phi(g)$ be defined by $$ P_\phi(g):= \sum _{\gamma \in \Gamma} \phi(\gamma g).$$ Theorem 6.1 in Borel's Automorphic Forms on $SL(2,\mathbb{R})$ says: If $\phi$ is $K$-finite on the right, then the series converges absolutely and locally uniformly, belongs to $L^1(\Gamma \backslash G)$, and represents an automorphic form for $\Gamma$; and if $\phi$ is $K$-finite on the left, then the series converges absolutely and is bounded on $G$.
Here is an example of an integrable function that is $K$-finite on both sides and $\mathfrak{Z}$-finite. For the rest of this post, let's switch to working in the unit disc model, instead of in the upper-half plane, using the transformation $T= \lbrace 1, -i; 1, i \rbrace$, which sends $i$ to $0$. So, from now on, when I write $G$, I mean $T.SL(2,\mathbb{R}).T^{-1} = SU(1,1)$; similarly for $K$ and $\Gamma$. Let $j(g,w):=cw+d$, where $c$ and $d$ are the bottom left and bottom right entries of the matrix $g$, respectively, and let $\varphi_n(w):=w^n$, with $w$ in the unit disc. Then $$ \phi_{m,n}(g) := j(g^{-1},0)^{-m} \varphi_n ( g^{-1}.0 ) \qquad (n,m \in \mathbb{Z}, \, n \geq 0, \, m \geq 4 )$$ has left $K$-type $m$ and right $K$-type $-m-2n$. In fact, $\phi_{m,n}$ is a basis element of a (holomorphic) discrete series representation of $G$ (acting on the right) in $L^2(G)$. And if $P_{\phi_{m,n}}(g)$ is not identically zero, then $P_{\phi_{m,n}}(g)$ is a basis element of a discrete series representation of $G$ (acting on the right) in $L^2(\Gamma \backslash G)$. The left and right $K$-types describe the characters by which $K$ acts on the left and the right, respectively; and saying that a function is $\mathfrak{Z}$-finite means that the function is annihilated by a non-constant polynomial in the Casimir operator. I could elaborate on the representation theory here, but I think this question is really about lattices.
Now denote $P_\phi(g)$ by $^L \! P_\phi(g)$, and define
$$ ^R \! P_{\phi}(g):= \sum _{\gamma \in \Gamma} \phi(g \gamma).$$
(The superscripts L and R stand for averaging over the left and right action of the lattice, respectively.) Then the mirror image of Theorem 6.1 in Borel's book holds.
Question. True or false: $$ ^L \! P_{\phi_{m,n}}(g) := \sum _{\gamma \in \Gamma} \phi_{m,n}(\gamma g) \not\equiv 0 \quad \Longleftrightarrow \quad ^R \! P_{\phi_{m,n}}(g) := \sum _{\gamma \in \Gamma} \phi_{m,n}(g \gamma) \not\equiv 0 $$ I think it should be true, but I cannot prove it. If one of the series is non-zero at some $g \in G$, shouldn't it be possible to find some $g' \in G$ where the other series is non-zero?
Observations and ideas. Note that $^L \! P_{\phi_{m,n}}(g) \in L^1(\Gamma \backslash G)$, with right $K$-type $-m-2n$, while $ ^R \! P_{\phi_{m,n}}(g) \in L^1(G / \Gamma)$, with left $K$-type $m$, so the situation is not entirely symmetrical.
Here's how the Poincaré series defined in the Background section are related to another kind of Poincaré series. Let $\varphi$ be a bounded holomorphic function in the unit disc, and let $m \geq 4$ be an integer. Then the Poincaré series $$ p_{m,\varphi}(w) := \sum _{\gamma \in \Gamma} j(\gamma,w)^{-m} \varphi (\gamma.w) $$ converges absolutely and locally uniformly and defines a holomorphic automorphic form for $\Gamma$. (This is Theorem 6.2 in Borel's book. By the way, these are not the same Poincaré series as the Poincaré series at infinity in e.g. Iwaniec's book on automorphic forms.) We have $$ j(g^{-1},0)^{-m} p_{m,\varphi_n}(g^{-1}.0) = \ ^R \! P_{\phi_{m,n}}(g)$$ where $\varphi_n$ is as in the Background section. On the other hand, I can't obtain any relationship between $p_{m,\varphi_n}$ and $^L \! P_{\phi_{m,n}}(g)$.
Anyway, I think that the answer to this question will have more to do with lattices than with automorphic forms. I'm aware of the relationship between dimensions of spaces of cusp forms and multiplicities of discrete series representations in $L^2(\Gamma \backslash G)$ or $L^2(G / \Gamma)$, but it doesn't help here. Could we use Borel's density theorem? ($\Gamma$ is Zariski-dense in $G$.) Could we use the fact that $G$ is unimodular? (A necessary condition for the existence of lattices.) Am I overthinking this?