Originally Posted by Mad Scientist

I assume you mean one output going positive and while a second output is going negative.
To do that just add a second IC and use ½ of it as an inverter and the other half as a buffer this will give two outputs that are 180 degrees out of phase with each other.

Note: One LM339 can be used in place of the two Lm393’s, as it is the quad version of the 393.

The 555 is a great little chip, with untold uses, but trying to do this with it is an exercise in learning how to tear your hear out.