The Sphinx is armed with a Spitfire and a elite BS of 14. It has two Heavy Flamethrowers for closer engagements, and an Explosive close combat weapon. Most of the time it will be using the Spitfire to take full advantage of the -6 to hit imposed by TO camo, but fast movement and climbing plus means it can get into so devastating positions to fire both templates of its Heavy Flamethrowers onto different units, covering a large portion of an enemy deployment zone.
Remember that the Explosive close combat weapon also means that it can damage destroy able scenery items or objectives, and in some scenarios such as Looting and Sabotaging, it even gains a bonus to do so.
Were it a heavy infantry the TAGs armor and BTS would be considered high, but it is among the lowest for a TAG in the game. It does have 3 points of structure but it is best to think of all of these as insurance for bad rolls rather than a perfect defense. Much more often, the Sphinx survives on its TO camouflage, either forcing Dodges or bad odds BS attacks from the opponent, or going back into camouflage state and climbing up a roof near the end of its active turn. More than almost any other TAG the sphinx values spending the last few orders of the active turn to get it out of harms way and it has ample abilities to do so with.
This model may use hidden deployment, meaning that you can note its position on the board and not place a model or marker at the start of the game and you should nearly always choose to do so. Keeping it off board means the opponent cannot target it until you are ready to bring the monster to bear. Do note that because of it's high points value, most opponents will realize that something is up and start to try and figure out what it missing. Knowing you have a Sphinx on the table and dealing with it are, however, different skill sets.
Climbing plus can allow you to deny cover in some situations by going to higher ground. You will lose cover yourself by doing so (as you can't claim partial cover while climbing) but it can still often be worth it.
Learning how the Stealth rule works is an important feature of the Sphinx. If you only perform a move+move then opponents without line of fire to the Sphinx cannot take an ARO.
Moving behind an enemy as a camouflage token gives them some difficult choices. They can chose to Change Facing, in which case you can shoot them without reprisal. Or they can "Hold ARO against camo" on your Sphinx. If they do the latter, you can continue moving, ideally more than 8" from the target and shoot them without the enemy even gaining an ARO.
The Sphinx fears anything that will deny it's camouflage. Troops with Multi-Spectral Visors or Flamethrowers can shoot it without penalty or burn off its camouflage and leave it significantly weaker. ARM 6 and 3 STR seem hard to deal with, but enough weight of fire will take the TAG out.
The Sphinx also is vulnerable to hacking. Though you cannot eject it's pilot, it can easily be locked down, or worse, Isolated with the hacking program Oblivion which renders it all but useless.
Finally, the Sphinx has a fine CC score and weapon but a Martial Artist can easily overcome these and take it out in close combat.
A Sphinx, like any TAG is a large portion of your list. In generic Combined Army you will want enough cheap orders to fuel it, but also make a back up plan of specialists, MSV troopes, and even another HI perhaps to support it. Troops with Infiltration are good in this function because even with a few orders spent retreating, the Sphinx is likely to be far forward. Q-Drones are also a nice option because of their dangerous AROs Because of it's cost it is also nice to bring a MedTech or other Engineer to repair it should the TAG go down, get isolated, or lose it's camouflage.
In the Shasvastii Sectorial cheap orders are hard to come by but a Speculo Killer pairs well with the Sphinx by taking out threats to it, or clearing a path for its advance by killing models or simply smoking the approach. Shrouded also play well with it. Two Ikadrons is almost mandatory.
In Onxy you can still bring a cheap link of Unidron Batroids but you will find yourself hungry for both points and Special Weapons Cost. Fraakta can be good for clearing the way for the Sphinx, and the Sphinx in turn is good for clearing the advance of a link to prime fighting ranges.
A Word on combat groups: One would be tempted to bring a full 10 orders for the Sphinx, but 9 orders and a 106 points missing makes your plan very obvious to an opponent. It can sometimes be worth bringing a cheap TO model or AD to help cover part of the missing points. Mine-Layer Shrouded also help with this by putting more tokens on the board so the opponent is less likely to notice how few models you have. The mines also help the Sphinx to retreat to a safe position to camouflage itself again.
When deploying the Sphinx worry less about who can see you initially and more about where your first moves will bring you. Even if you do get discovered from across the board, you can usually plan to be out of all Line of Fire by the end of your second move skill. And you do want to be Move + Moving the first order in which you use the Sphinx because it's weapons have a very short optimal range. Look at the enemy side of the table and figure out how best to get your killer across the board in the minimum number of orders and ARO. Climbing is often a blind spot for most players. Use this to your advantage.
Another note on Climbing Plus: You can deploy clinging to a wall with this skill which can allow you to be in some really weird places. You do not have Super Jump so you will still have to climb down, but enemies often look for the TAG shaped hole in scenery when guessing at where you deployed and don't think about elevation.
Active Turn Role
The Sphinx is meant to kill line infantry. It lacks the hitting power to take out heavier armor and is best served when it can abuse its camo against a foe without a visor or superior stats or weapons. If Shasvastii is the bully faction, this unit is the king. Kill as many line troops, remotes, doctor/engineers, and other weak troops as you can. Lieutenant hunting should be left to other units, because unless your foe has made a serious mistake, there will be too many ARO watching the approach. Instead pick off isolated troopers. Don't risk shooting it out with a visor unless you a very favored. If the Sphinx only kills 3 of 15 enemies each turn, but manages to get to complete safety in doing so, it has done it's job.
Because it is so expensive you may be tempted to go first whenever using the Sphinx. However if you can protect your forces it is sometimes better to take deployment and wait for your opponent to string themselves out on their first turn to give you holes to exploit.
The Sphinx can act as a specialist as well thanks to it's Scindron Auxiliary Remote. This is order intensive and leaves the TAG vulnerable. If you have to do it, be sure to have enough orders to get the Scindron back to the TAG or to reset with the TAG and abandon the drone and still get back to camouflage.
The Sphinx should avoid ARO whenever possible. There are niche situations, say when an entire link team walks in front of your flamethrowers, or a 1 wound Lieutenant strolls past in a good range band. If you really need to hold ground, the Sphinx can also go into Suppressing Fire and be hard to deal with due to its TO modifiers, but be sure that you have a very short field of fire or you are likely to get a Missile from outside of 24" for your efforts.