3.5 tried to simplify facing rules, and in the process... well, let's just say that I think 3.5 was entirely a move in the wrong direction.
First of all, facing in combat in most cases is irrelevant regardless of the space that a creature fits into. All 3.0e's 'no facing' rules says is that since the combat round is six seconds, all creatures are assumed to not have a constant facing during combat. They are not forced to limit their facing to fit into the arbitrary turn based system, but are assumed to be dynamically adjusting it. Thus, except as provided by the flanking rules, creatures don't have a 'back' in the way that they did in 1e were facing was of great importance.
So it really doesn't matter if a horse as a 5x10 facing. In 3e, it can freely attack and defend itself in all directions nonetheless, and it can move in all directions freely. The space it occupies only matters for how you 'rest' the character between actions. 3.5's simplification was trying to avoid the problem of a long creature gaining extra movement by adjusting its orientation so that it was actually moving around its axis without the need to call out the case specifically. Fine, but it loses tons of verisimilitude in doing so, and often becomes ridiculous - a long snake cannot move down a narrow corridor in 3.5 unless you basically ignore the rules.
For my part, I still use something like the 3.0e rules. In general, I have no need to call out explicit facing. However, my house rules for flight DO require that an airborne creature maintain facing since most airborne creatures can't adjust facing freely. That is, most flying creatures cannot hover and spin in place. Those that can - say an air elemental - don't have to track facing since they are as stable in flight as most land creatures are on the ground.
I generally don't have a fighting chariots in my game, as they are obsolete technology in the setting. Also, generally when I run chase scenes I do so abstractly, since the turn based nature of D&D combat is ill suited to chases. However, if combat on vehicles - whether ships or chariots or wagons - was going to be something I was going to run, I think I'd feel the need to give such vehicles a maneuverability class much as I did for flying creatures. This would require the vehicle to maintain a facing and to update that facing only after moving forward a number of squares. I would be far easier to track this sort of thing than it is to track flying, since you'd only need to track facing and not facing and elevation. Aerial combats basically are going to eat up an entire session due to sheer complexity and difficulty in imagining them. But I doubt a chariot combat is going to play out any slower than standard D&D combat in 3e.
For your boar chariot I'd require that it only be able to change facing by 45 degrees for every 4 spaces of motion in the direction of its facing. It advantages would be:
a) More stable archery platform than a horse, at least on flat terrain.
b) Capable of a powerful charge and trample attack by moving over medium sized or smaller creatures.
c) Capable of a blade attack on all adjacent creatures as it moves.
d) Provides partial cover from attacks to the riders. However, the driver himself probably loses his dexterity bonus as he must spend his time controlling the team and he cannot himself make attacks while doing so. Most combat chariots featured a two man team of driver and archer, or else used the chariot in a fashion somewhat similar to an armored personnel carrier to deliver shock troops to critical positions in the battlefield. This chariot seems to do neither, and is focused entirely as chariot in its role as means of overrunning light infantry by trampling/scything them. Note that this style of chariot, as popularized by the Hittites, did not last long as a militarily significant weapon system since it dies hard to disciplined infantry. Basically, in D&D terms, anyone that hold their action and has a pole arm can carve this chariot up and there is nothing the driver can do about it - move to the side as a declared action and take the AoO as it goes past you. Much like a flier with poor maneuverability trying to fight an air elemental, after the surprise round (if any) the poor flier never gets to make an attack against someone using good tactics.
Make sure you make the chariot itself reasonably durable - hardness 8 and say 48-64 hit points is probably reasonable. Treat the boars and driver as separate targets for all other purposes than movement.
You could of course make the rules as complex as you desire, with animal handling checks to avoid crashes if moving through difficult terrain or attempting fast turns.
