Catalyzes the hydrolysis of S-inosyl-L-homocysteine (SIH) to L-homocysteine (Hcy) and inosine. Likely functions in a S-adenosyl-L-methionine (SAM) recycling pathway from S-adenosyl-L-homocysteine (SAH) produced from SAM-dependent methylation reactions. Can also catalyze the reverse reaction in vitro, i.e. the synthesis of SIH from Hcy and inosine. Is specific for SIH and inosine as it is unable to either hydrolyze SAH or synthesize SAH from adenosine and Hcy. H2O + S-inosyl-L-homocysteine = inosine + L-homocysteine Binds 1 NAD(+) per subunit. kcat is 0.83 sec(-1) for the synthesis of SIH from inosine and L-homocysteine. kcat is 0.42 sec(-1) for the hydrolysis of SIH (at pH 7.0 and 70 degrees Celsius). Optimum pH is 7.0-7.5 and pH 9.6 for the reaction in the direction of SIH synthesis. Shows no activity at either pH 6.5 or pH 11.5 in the synthetic direction. The activity of SIHH in the hydrolysis direction shows essentially the same activity over the pH range 6.5-11.5. Optimum temperature is about 70 degrees Celsius. Amino-acid biosynthesis; S-adenosyl-L-methionine biosynthesis. Exists both as a homotetramer and a homodimer, in a 4:1 ratio. SAH is a product of SAM methyltransferases and is known to be a feedback inhibitor of these enzymes. As a result of this inhibition, organisms have evolved efficient enzymes to metabolize SAH via different pathways. The pathway found in methanogens differs from the canonical pathway, it uses the deamination of S-adenosyl-L-homocysteine to form S-inosyl-L-homocysteine for the regeneration of SAM from S-adenosyl-L-homocysteine. Belongs to the adenosylhomocysteinase family.