Just to add to TVP's and Bilge's fine postings:
I would interpret the 32-38 as an additional requirement on the steel. As TVP pointed out, the standard callout for 8620H (8620 with guaranteed hardenability) is 27 to 41 at the 4/16" position on the Jominy bar. (Hallen, your interpretation was correct.) Hence to get a tighter 32-38 you need to get the steel company to refine the composition & grain size, or buy what a distributor has that meets the requirement.
ASTM A 304 is the ASTM analog of J1268, and the Jominy hardneability test specifications are ASTM A 255 and SAE J406. Of interest might be the "restricted hardenability" bar specifications, ASTM A 914 and SAE J1868, which has a Jominy hardenability of 30 to 38 at the 4/16" position for 8620RH. Not quite, but closer to your mark...
Good luck!
Lee