"Demons To Some; Angels To Others" - The Order of the Gash for Old School.

We'll stick with Cenobites for convenience, but they're called other things by those outside the Order: Cold Ones, Flesh Mechanics, Inquisitors, Surgeons, Theologians, Hell Priests, Clinicians, Engineers, Keepers, Sensates etc. 

Cenobite/ Clive Barker/ 1986.

You know what they are and what they do, right?

"It is not hands that call us... it is desire".

They come when called and will not willingly go back unaccompanied. They can be bargained with, but are instinctively proprietorial. While pain-and-pleasure indivisible is their meat and drink, they are not compulsive missionaries to the innocent/ignorant.

Solving the Puzzle opens the Schism, allowing d6 Cenobites to pass through, but it is the obsession for a solution or desire to contact the Order that summons them.

You must figuratively or literally...

1. open the Box
2. count the Stones
3. walk the Labyrinth
4. cross the Threshold
5. escape the Maze
6. set the Clock
7. debug the Program
8. crack the Code
9. play the Music
10. cast the Spell
11. grow the Tree
12. deliver the Child
13. honour the Mother/Father
14. slay the Dragon/ defeat the Minotaur
15. get the Joke
16. draw down the Moon
17. navigate the River
18. transcend the Flesh
19. raise the Cathedral
20. plumb the Depths

You can be tricked into solving the Puzzle on behalf of another, and the Order is willing to educate the innocent even though they prefer to reward the guilty. Your desire to contact the Order can be subconscious.

The Schism allows traffic both ways.

What is it Wearing?

Cenobite/ Clive Barker/ 1986.

1. Gory Skin Suit (Unarmoured)
2. Naked Marmoreal Perfection (Unarmoured)
3. Butcher’s Apron and Gauntlets (Leather)
4. Full S&M (Leather)
5. Full S&M with Extreme Implants (Scale)
6. Hooked and Spiked Chains (Chain)

Bod mod, ornament, masks, jewellery, helmets, gloves, boots, uniforms, cosplay etc., to taste.

What is it Wielding?

Cenobite/ Clive Barker/ 1986.

1. Antique Hand Weapon (sword, mace, warhammer etc.)
2. Notched Heavy Cutting Blade
3. Nothing, Apparently
4. Tools of a Trade (surgeon, butcher, whaler, taxidermist, lepidopterist etc.)
5. Whip/ Scourge
6. Brutal Flail

That a Cenobite wouldn't be carrying a wicked knife somewhere about its person is almost unthinkable, but a spike, hook, blade, strangling line or cudgel are all startlingly easily improvised from everyday objects and body parts. 

Stats/Mechanics.

Cenobites are ascended/corrupted mortals, but are neither demonic nor undead despite their strong affinities.

Some of my choices for suitable reskins:

1. Bugbear
2. Shadow
3. Troglodyte
4. Golem, Wood
5. Mummy
6. Troll

Early Cenobite/ Clive Barker/ 1986.

Per combat round, one attack can be reskinned as an animated hooked/spiked chain. In addition to damage, this will constrict/entangle/grapple as a whip, tentacle or similar.

If they are not already immune to normal weapons, consider treating them as Invulnerable Monsters anyway - particularly if you're using them for horror purposes.

Cenobite/ Clive Barker/ 1986.

Personality:

1. Absent-minded Professor/ Eager Physician
2. Pompous Martinet/ Avant Garde Vivisectionist
3. Jaded Aesthete/ Jovial Sadomasochist
4. Aching to Fill the Hollow/ Transdimensional Jobsworth
5. True Believer/ Imaginative Taxonomist
6. Brutal Sycophant/ Fallen Avenger

Roll 2d6; read one as odds/evens.

Generally, only one Cenobite in a party will have a personality distinct from the others.and will probably but not always be both the leader and most powerful. All subservient Cenobites will be either/or of result 4.

Alignment: Lawful Evil Neutral.

The Order styles/ claims itself to be beyond human morality and outside the mainstream cosmology of your setting, beholden only to themselves. 

Protection from evil and the like works against them because they're summoned monsters.

Intelligence: Human scale, though with the advantage of forbidden knowledge and weird experience across an immortal existence.

Acedia.

Having explored the furthest regions of experience, some Cenobites get bored with being Cenobites (double ones on 2d6, either no more than one in a group or it's all of them) and behave uncharacteristically: 

• showing mercy
• studying and practising magic
• planar adventuring
• attempting to kill God
• attempting the dethrone the Adversary
• generally upsetting the balance of the cosmos
• indirectly opposing the Order
• falling in love
• offering to trade places with mortals
• remembering their lost humanity
• seeking glory or redemption
• taking up mundane hobbies
• directly opposing the Order
• trying to divide and conquer the Order
• seeking Ascension
• slumming it as an earthbound haunting spirit

Cenobite/ Clive Barker/ 1986.

Don't Just Use Kytons.

D&D adjacent, the most obvious choice is to reskin a Kyton, as the Chain Devil is already an off-brand Cenobite anyway. 

This isn't meant as absolute proscription.

What Was Left Out/ Further Reading.

There's a lot of conflicting canon out there. How much lore do you want?

Compare and contrast the Cenobites in The Hellbound Heart, the first two Hellraiser movies, and The Scarlet Gospels - Clive Barker himself takes them from weird sort-of-sexy mystery, through Lovecraft-adjacent menace, to rather conventional demons in an apparently Judeo-Christian cosmology. 

Plus, there are some minor references across his other works, and prototypes in The Books of Blood.

Then consider the later films, the unrealised projects, the comic books (which in particular extend them into history and myth), and pseudo-cameos in Extreme Ghostbusters, Dark City, Event Horizon and The Cabin in the Woods (maybe The Void, too).

There're at least two Hellraiser Wikis on Fandom.

Not more than passing familiar with either, but The Machine Orthodoxy (MTG) and the Kyton monster type (Pathfinder) set their tents out on Cenobite territory. 

Via EN World - Hellraiser for d20 Call of Cthulhu (so 3e compatible, I suppose).

On Tumblr, Hellbound for 5e D&D at The 5th Edition Foundry.