infinite impulse response - complementary IIR filters

I would like a pair of complementary IIR filters (lowpass/highpass). By complementary, I mean, when the output from the two filters is summed, the original signal is recovered. I thought I could build such pairs with butterworth filters but using a little math, I discovered only 1st order filters were complementary. I thought I'd done this before, but I'm forgetting how.

Is something wrong with my math? Is there an easy solution I'm forgetting about?

Thanks!

Answer

Juancho's answer is sort of right, however there is one problem: The complimentary filter of a low pass is generally NOT a high pass filter, at least not in the sense that you are looking for. For example the one's compliment of a 4th order Butterworth low pass does not look like a 4th order high pass filter at all. It has about only half the steepness, reaches a maximum gain of ca. +6 dB below the crossover frequency and than slowly approaches unity gain above the cross over frequency.

The only matching high and low pass filters that sum to unity are first order filters. You can however find matching higher order filters that sum to unity gain so that the overall transfer function of the sum is an all pass filter. These are odd order Butterworth filters and even order Linkwitz Riley filters.