Here's a list of the uniquely different print profiles available, sorted by ink limit, along with their driver names. In general, higher ink limits mean wider print gamuts. To choose a profile, then, find the highest ink limit that has a profile that reasonably matches your paper type, and do a calibration/ICCe series. If you find mottling (spots) or ink pooling in the darker colours on the ICC print, choose a similar profile with a lower limit and try again.
N.B. What I've gathered from the HP document makes me believe that there is no benefit to choosing an HP branded paper type from the list if that paper's profile is also available as an unbranded paper type. The document doesn't state or imply that colour mixing is different if all other parameters are identical, so I'm pretty sure this is the case. Contrary evidence is welcome.
Bi - 16 pass bidirectional printing
Borderless - Borderless option available (omission means unavailable)
Gloss - Gloss enhancer is usable (omission means not usable)
High - high carriage (omission means low carriage)
Matte - Matte black used.
Nocut - Cutter cannot be used.
Photo - Photo black used.
Quad - Both Matte and Photo black used.
Uni - 8 pass unidirectional printing
Ink Limit 20
Coated / Plain - High, Matte, Uni
Ink Limit 22
Coated / Coated - High, Matte, Uni
Ink Limit 30
Photo / Photo Matte Paper - High, Matte, Bi
Proofing / Proofing Matte - High, Matte, Uni
Fine Art / HP Collector Satin Canvas - High, Matte, Bi, Nocut
Ink Limit 36
Photo / Photo Gloss Paper - Gloss, Borderless, Photo, Bi
Proofing / Proofing Gloss - Gloss, Borderless, High, Photo, Uni
Ink Limit 42
Fine Art / Canvas - High, Matte, Bi, Nocut
Fine Art / Fine Art Paper - High, Matte, Bi
Ink Limit 46
Photo / Glossy Paper High Ink - Gloss, Borderless, Photo, Bi
Proofing / Glossy Paper High Ink - Gloss, Borderless, Photo, Uni
Fine Art / Fine Art Media Pearl - Gloss, Borderless, High, Photo, Bi
Ink Limit 48
Coated / Superheavyweight - High, Photo, Uni
Ink Limit 60
Fine Art / Fine Art > 250g/m2 - High, Quad, Bi
Here's the same list, organized by black used.
Photo Inkset (Gloss, Borderless)
Limit 48 - Coated / Superheavyweight (high, uni)
Limit 46 - Photo / Glossy Paper Hi Ink (bi)
Limit 46 - Proofing / Glossy Paper Hi Ink (uni)
Limit 46 - Fine Art / Fine Art Media Pearl (high, bi)
Limit 36 - Photo / Photo Gloss (bi)
Limit 36 - Proofing / Proofing Gloss (hi, uni)
Limit 42 - Fine Art / Canvas (bi, nocut)
Limit 42 - Fine Art / Fine Art Paper (bi)
Limit 30 - Photo / Photo Matte (high, bi)
Limit 30 - Proofing / Proofing Matte (high, uni)
Limit 30 - Fine Art / HP Collector Satin Canvas (high, bi, nocut)
Limit 22 - Coated / Coated (high, uni)
Limit 20 - Coated / Plain (high, uni)
Limit 60 - Fine Art / Fine Art > 250gsm (quad, bi)