Causal unstable system turn into stable anticausal?
An unstable system can be stablized by all-pass decomposition for exampe, but simply constructing the anti-causal impulse respone extension will not provide the same output; so I think it's not possible, but I'm not rigorous at this point. Following just shows why.

Assume that your unstable and causal IIR filter has the impulse response $h_+[n]$ and by evaluating it into $ n &lt; 0$ domain you construct a stable but anti-causal impulse response denoted by $h_{-}[n]$.

Your true output given by the unstable filtering is 
$$ y[n] = h_{+}[n] \star x[n]$$

Now you construct alternate output from a reversed input and the anti-causal filter as
$$ y_{-}[n] = h_{-}[n] \star x[-n] $$

And by reversing $y_{-}[n]$ , do you get $y[n]$ ? 

No as seen:

$$y_{-}[-n] = h_{-}[-n] \star x[n] \neq h_+[n] \star x[n] $$

that's to say unless $h_{-}[-n] = h_+[n]$ , they wont be the same. And indeed the reversed anti-causal impuse response will not be equal to the causal one, since one is stable and the other is not.