High Pass Filter Inverse
For specific cases, you can. For others, you can't. So we can't answer regarding your specific filter without knowing its circuit or transfer function.

For an example where it's possible, look at pre-emphasis and de-emphasis networks, used to boost a range of frequencies before a noisy transmission system, and reduce them (also reducing the noise) at the receiver.

For an example where it isn't, look at your transfer function : is there any frequency - say, DC for a HP filter - where the output is zero? ( a so-called "zero" in the transfer function) If so, you cannot design a matching filter that covers that frequency. Having eliminated DC, you must use some other technique such as a "black level clamp" in analog video to restore it.

However you may be able to design a matching filter for a spectral range that excludes the zero - such as a frequency range down to 20Hz, considered "good enough" for most audio purposes.

With more complex transfer functions involving second order filter sections, you may be able to undo the amplitude response but not the phase response. 

If you need to cancel changes to the phase response, the best you can do is add a "group delay equaliser" usually implemented as an "all pass filter". This does not undo the phase changes (you would need to wind time backwards for that!) but adds further delays such that the entire signal is delayed by a constant time. 

A constant time delay appears in phase measurements as a phase shift linearly increasing with frequency (30 degrees at 1 kHz is the same as 60 degrees at 2 kHz) and is often called a "linear phase" system.