It's possible to create a supply that can sweep from +15V to -15V but the way you've described won't work for loads above 1mA at the most.
The trouble is that these regulators can only supply current, they can't accept it. In other words an LM317 set to +10V can push current into a load connected to 0V but it can't accept current if the load is connected to +15V.
The correct way to do this is to build a push-pull amplifier (e.g. an audio amplifier) and supply the input with a DC reference voltage ranging from +15V to -15V. Providing the amplifier is all DC coupled there shouldn't be any problems.
How do you get the reference voltage?
Easy, use a centre tapped transformer with the usual bridge rectifier and connect a 15V zener, in series with a suitable resistor, from the positive and negative to 0V.