Explanation of bond angles in the aluminium chloride dimer
The bonding in aluminum chloride ought to be considered largely ionic. The electronegativity of $\ce{Al}$ is $1.47$ vs $2.83$ for $\ce{Cl}$ (for comparison: $\ce{C}$ $2.50$; $\ce{Mg}$ $1.23$; $\ce{Na}$ $1.01$). Molecular orbitals and covalent bonding can be added for a fuller picture of electron densities.

Aluminum chloride solid (Melting point $\pu{193^\circ C}$, under pressure) has a cubic close-packed array of chlorine atoms with only $2/3$ of the octahedral holes between every other pair of chlorine planes occupied by the aluminum atoms. At low temperatures in the vapor state, aluminum chloride is a tetrahedrally coordinated dimer, and at very high temperatures it is a trigonal monomer. So aluminum can easily have $3$, $4$ or $6$ coordination. Whatever directionality the bonds have from hybridization of the $\ce{3s}$ and $\ce{3p}$ orbitals is mixed with plain electrostatic (ionic) attraction.

Imagine putting just the 6 chlorine atoms together. There will be two tetrahedral holes for aluminum (see image below). The radius ratio 
$$\frac{\ce{Al^3+}}{\ce{Cl-}} = \frac{\pu{50 pm}}{\pu{181 pm}} = 0.276$$ 
is well within the stability range ($0.225 - 0.414$) for tetrahedral coordination. So you might expect all the $\ce{Cl-Al-Cl}$ bond angles to be $109^\circ$. 

But while the void for the aluminum is big enough, the distance between the internal chlorine atoms would be only $\pu{300 pm}$ in this configuration; the chlorine atoms are squeezed (ionic radius $\pu{181 pm}$). In addition, the $\ce{Al-Al}$ distance is only $\pu{250 pm}$ and the $\ce{Al^3+\bond{-}Al^3+}$ repulsion would be extreme. (In the solid, the octahedral holes are farther apart) So both aluminum atoms and the internal chlorine atoms are pushed apart, consistent with imagining more $\ce{p}$ character in the central bonds and therefore less $\ce{p}$ character in the external chlorine atoms, so $\ce{Al}$ goes to $\ce{sp^2}$ (or nearly so). The internal $\ce{Cl-Al-Cl}$ bond angle is even less than $90^\circ$ (less than you would expect for two $\ce{p}$ orbitals), so &pi; bonding needs to be considered.

Aluminum chloride is a strong Lewis acid. Its solubility in organic solvents is due to formation of complexes. It is not soluble in the way that carbon tetrachloride or hexachloroethane are soluble. The salt-like (ionic) character of solid aluminium chloride is still present in $\ce{Al2Cl6}$; the aluminum atoms are surrounded by a cocoon of chlorides. Solid aluminum chloride can be ground into particles and nano particles; $\ce{Al2Cl6}$ is like a pico particle of salt.