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Backswing (Ex): At 7th level, when a two-handed fighter makes a full attack with a two-handed weapon, he adds double his Strength bonus on damage rolls for all attacks after the first.
All double weapons qualify as Two-handed weapons. So the question is while using both sides of a double weapon would the two-handed fighter ability Backswing apply?
A double weapon is a type of two-handed weapon; every double weapon in the game is listed in the appropriate Two-Handed Weapon table. The rules for double weapons state that, when you use both ends of one in conjunction with the Two-Weapon Fighting option, the double weapon is treated as if the two ends of the weapon were a one-handed and light weapon, respectively, for the purpose of TWF attack penalties. Nothing defines them as such for any other purpose, so aside from deciding your attack penalties from Two-Weapon Fighting, double weapons are two-handed weapons.
Therefore, double weapons are two-handed weapons, and therefore benefit from Backswing, no matter how they are used.
However, please note that though this is technically true even when you use Two-Weapon Fighting, this is widely, if not universally, regarded as a rules oversight. The rules also specifically mention using only one end as a two-handed weapon, which would be a meaningless option if you could use both ends as two-handed weapons. Thus, most tables treat double weapons as if the ends were separate weapons for all purposes, rather than just for TWF attack penalties, while you use Two-Weapon Fighting.
Aside from my own games, I have only played under exactly one DM who allowed double weapons to function as two-handed weapons for all purposes (other than TWF attack penalties) even when used for TWF. In addition to Backswing, this implies that you get 1½×Str to damage with both ends, and you get the improved Power Attack damage bonus that two-handers get with both ends. Since every double weapon in the game is quite weak, I think this is appropriate benefit for them.
A double weapon takes two hands to use, but only counts as a one-handed and light weapon when you are attacking with both ends. In order to use a double weapon to make a two-handed attack, you have to only use one end at a time:
The character can also choose to use a double weapon two-handed, attacking with only one end of it.
If you're using both ends, you can't make a two-handed attack.
