How to design an all pass type filter with inverted phase response?
Sort of. The inverse of an allpass filter is also an allpass filter and can simply be calculated by time flipping the impulse response. However that makes the filter non-causal. So you need to add bulk delay to make it causal again. As long as you don't need to do real time processing, that's typically not a problem. 

EDIT: add a specific example. The key is to do the filtering in the time domain. You need to truncate the IIR response to an FIR response using "enough" samples. 



% create an arbitrary allpass filter
b = [-.5 1]; a = [1 -.5];

% calculate the impulse response
n = 128; d0 = zeros(n,1); d0(1)= 1;
h1 = filter(b,a,d0);

% time flip and shift by 1 to keep the first sample in the same spot
h2 = circshift(flipud(h1),1);

% plot frequency responses
fh1 = fft(h1);
fh2 = fft(h2);
phases = unwrap([angle(fh1(1:n/2+1,:)) angle(fh2(1:n/2+1,:))]);
plot(phases);