There are probably too many such $(M,g,I_+,I_-)$ to really expect a 'classification'.  

For instance, consider the case when a complex manifold $(M,I_+)$ has real dimension $4$, and the $I_+$-holomorphic tangent bundle $T'M$ splits as the sum of two holomorphic line subbundles $T'M = L_1\oplus L_2$.  In this case, one can define an $I_-$ by reversing $I_+$ on the line bundle $L_2$.  Now let $\Omega_i$ for $i=1,2$ be a smooth real (1,1)-form of complex rank $1$ whose kernel is $L_{3-i}$ and is such that $\Omega_i$ restricts to $L_i$ to be a positive $(1,1)$-form (with respect to $I_+$).  Let $g$ be the Hermitian metric (with respect to $I_+$) that is associated to the positive $(1,1)$-form $\Omega_+ = \Omega_1 + \Omega_2$ (with respect to $I_+$).  Then the $2$-form $\Omega_- = \Omega_1 - \Omega_2$ is a positive $(1,1)$-form with respect to $I_-$ and it belongs to the same associated metric $g$. 

It's now easy to check that the two Lee forms satisfy $\theta_+=\theta_-$.  Note that replacing $\Omega_1$ and $\Omega_2$ by $f_1\Omega_1$ and $f_2\Omega_2$ where the $f_i$ are any positive (smooth) functions on $M$ will give other examples with equal Lee forms.

It's not hard to show that, in dimension $4$ this Ansatz the only way that such examples can arise.  In higher dimensions, I expect that the analysis is more complicated, but, again, there will be too many examples to really classify because having equal Lee forms is a very underdetermined set of PDE on the data $(M,g,I_+,I_-)$.