HILIC covers several types of stationary phases including bare silica, amine, diol, and others. I saw an article claiming cyano was run as HILIC although my experience seems to run it as reverse phase as aqueous systems.
Amine and diol generally run as normal phase with organic solvents, may run as normal phase from 0 to 50% water, and generally run as reverse phase with more than 50% water.
Something interesting that can be done with silica based C18, but not styrene divinyl benzene, is non-aqueous reverse phase where one uses methanol or acetonitrile instead of water, and a less polar solvent to elute the compound. The polymeric reverse phases swell in these solvents and generate high back pressure.
The links below are some of my work for flash columns, but should apply also to HPLC.
Teledyne ISCO doesn't sell HPLC columns anymore. I'd look to Waters and Agilent first. They have many application notes too, though I haven't seen any method development app notes- the links above partially fill that hole.
Hypercarb column from Thermo Fisher Scientific. is graphite carbon based column having unique separation, but this column takes long time to equilibrium. you can find application from the following paper: Journal of Pharmaceutical and Biomedical Analysis 86 (2013) 198–203.