Circuit that adds offset if input voltage is negative and subtracts offset if input is positive
Any dc offset (that's the word) that you want to remove can be simply done with a high pass RC filter: -



If your actual signal has a dc value that you want to pass through the circuit then you could use an instrumentation amplifier because it has a pin that your can use to set-up a DC offset that counters the DC offset you want to remove: -



You pretty much drive it like an op-amp but pin 5 (reference) can be set-up with an offset voltage that biases your real signal in one direction or the other.

If the offset is always the same magnitude and it's polarity is dependent on the polarity of the input then you can use the same idea to invert or not invert that offset depending on the polarity of the input. Use of a precision rectifier would help in determining the polarity of the signal and this can be used to switch either the positive offset or the negative offset to pin 5 of the instrumentaion amplifer. Analogue switches are probably a good choice for the selection of the offset source.