How to improve symmetry of +/-1 gain circuit?
Using a MOSFET for Q1 may be easier.  

A larger value of R3 will allow the transistor to have more effect. Other stuff will need to be changed to suit.

Massively overdriving a BJT so that it has a forced beta of say 0.1 or even less allows a very low saturation voltage. Long ago I had a BJT switching a resistor and the junction needed to be ~ 0. Any Vsat added to the signal at the other end of the resistor. Driving the base for 10 x as much current (or more) as the collector greatly improved the result. 

I don't now recall the forced beta value I used - may have been 10x - may have been 50x. As Icollector was low the actual base current was not vast. 

If you can find a whisper (or more) of negative bias lying around you may be able to mix some in. Using an opamp with feed back and the ability to pull to or below ground would be useful.

Whether you can do the following depends on circumstances and desired realness [tm] of circuit. -  returning negative lead from local ground to psu via a diode gives you a point one diode drop below ground on the cathode side of the diode. Can be very useful. 
eg single supply opamps with their ground pin returned to -Vbe below local ground can then  truly genuinely swing their Vout to local ground rather than to N mV above it.