What is the function of the connection between the two halves of the thalamus? What kind of direct exchange do we have between the two halves of the thalamus, whether going through the thalamic adhesion or not? Do we know what parts of left and right thalamus are connected and what kind of information they exchange?
Short answer
The function of the thalamic adhesion in humans is unknown.

Background
The two halves of the thalamus are separated by the third ventricle. In about 80% of people, the thalami are connected via the thalamic adhesion (massa intermedia, Fig. 1), which mainly contains cell bodies, but also commissural fiber systems that connect some thalamic nuclei (Satheesha &amp; Soumya, 2010; Sen et al., 2005). Because the structure is absent in about 20% of the people, it is not a critical structure. The significance and consequences of its presence or absence is not known in humans (Sen et al., 2005). 

It is known that there are sex-differences between the presence of the structure. It is present in 78% of females and 68% males (Allen &amp; Gorski, 1991) and it is about 50% bigger in females, when present. In schizophrenics, female patients were shown to have a significantly higher incidence of absent massa intermedia (33%) than their healthy controls (14%), whereas male patients showed no such difference (Nopoulus et al., 2001).    

In rhesus monkeys, ablation of the structure has been shown to lead to ipsilateral motor deficits (Lumley, 1972), but that has likely no relation to the commmissural tracts in question.  


Fig. 1. Dissection showing the massa Intermedia connecting the two thalami through the third ventricle. source: Wikipedia

References
- Allen &amp; Gorski, J Comp Neurol (1991); 312(1):97-104
- Lumley, Brain (1972); 95(2): 347-56
- Nopoulus et al., Schizophr Res (2001); 48(2-3): 177-85
- Satheesha &amp; Soumya, Int J Anatomic Var (2010); 3: 174–5
- Sen et al., Neuroanatomy (2005); eISSN 1303-1775 • pISSN 1303-1783