SAE J1268 Hardenability Bands for Carbon and Alloy H Steels gives the following range for alloy 8620H:
                     
"J" distance   HRC  
sixteenths
of an inch   max   min

    4            41    27

So, the range that you have been given of 32-38 HRC is a subset of the requirement for SAE J1268.  You can obtain a copy of this standard from SAE at http://www.sae.org .

A post-script to the previous response.

These days the Jominy curve for a steel can be calculated from knowledge of the steel's actual (not spec'ed) composition and its austenitic grain size.  See the current ASTM spec for the technique.

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