Consider the simplest possible case in which the time reversal operator $\hat{\mathrm{T}}$ is given by the operation of complex conjugation $\hat{\mathrm{K}}$.
We can view $\mathrm{T}$ is an anti-untary operator on the Hilbert space of our quantum system. In particular, we can determine its action on the momentum operator $\hat{\mathbf{p}}$ as
$$ \hat{\mathrm{T}} \hat{\mathbf{p}} \hat{\mathrm{T}}^{-1} = - \hat{\mathbf{p}}. \hspace{2cm}(1) $$
Question: How can we describe this operation within the phase space formulation of quantum mechanics? Can the form of this operation be derived from the Wigner transformation?
Some ideas: In this case, all non-commutativity of quantum mechanics is absorbed into the Moyal product, denoted by the binary $\star$ operation. The relation above will be transformed to its phase-space counterpart
$$ \mathrm{T} \star \mathbf{p} \star \mathrm{T}^{-1} = - \mathbf{p}. \hspace{2cm}(2) $$
It seems to me that the phase-space representation $\mathrm{T}$ of the operator $\hat{\mathrm{T}}$ should be rather non-trivial. My suspicion is that it will still contain the operation of complex conjugation (because the relations above should also hold when we consider the canonical momentum in electromagnetic fields).
The difficulty arises because $\mathbf{p}$ is now a real $c$-number and not a complex operator. Thus, $\mathrm{T}$ must have a dependence on position and momentum.
Conjecture: Based on these observations, I would conjecture that
$$ \mathrm{T} = \mathcal{U}(\mathbf{x},\mathbf{p})\mathrm{K}, $$
where $\mathcal{U}$ is a yet to be determined $\star$-unitary function on phase space. One thing which can easily be done is a translation $t$ by $\mathbf{p}'$, i.e.,
$$ t(\mathbf{p'})\star \mathbf{p} \star t(\mathbf{p'})^{-1} = \mathbf{p} +\mathbf{p}'. $$
This operation is easily defined. After we take the limit $\mathbf{p'}\to -2 \mathbf{p}$ we get the desired result. Can we incorporate this into the conjectured form?
EDIT 1: Following the discussion in the comment section we could interpret the statement (2) from above as an integral operator, let's call it $\tilde{\mathrm{T}}$ which acts via the functional
$$ \tilde{\mathrm{T}}[f(\mathbf{x},\mathbf{p})] = \int \text{d}\mathbf{x}'\text{d}\mathbf{p}' \delta(\mathbf{x}-\mathbf{x}')\delta(\mathbf{p}+\mathbf{p}') f(\mathbf{x}',\mathbf{p}') $$
Because of its operational definition, it is clear that this coincides with the Wigner transformation $\mathcal{W}$ of Eq. (1). It remains to show, that we can write this in terms of a $\star$-product, i.e.
$$ \tilde{\mathrm{T}}[f(\mathbf{p})] = \mathcal{W} [ \hat{\mathrm{T}} f(\hat{\mathbf{p}}) \hat{\mathrm{T}}^{-1} ] \overset{\text{to show}}{=} \mathrm{T} \star f( \mathbf{p} ) \star \mathrm{T}^{-1}. $$
EDIT 2: I might have found some additional constraints. If we assume that $\mathrm{T} \star \mathrm{T}^{-1} = \mathrm{T}^{-1} \star \mathrm{T} = 1 $, Eq. (2) can be used to derive
$$ \left\lbrace \mathrm{T} ~\overset{\star}{,} ~\mathbf{p} \right\rbrace= 0, $$
which is only fulfilled for $\mathrm{T} = 0$ which is not a solution to the original problem. This means $\mathrm{T}^{-1}$ cannot coincide with the $\star$-inverse of $\mathrm{T}$.