Place the eggs in a pot, cover with water and bring to full boil. Turn off the heat, cover the pot and let stand for 10 minutes.
Drain then crack the eggs by shaking the pot and running them under cold water until cool enough to handle.
Peel the eggs and halve them lengthwise or trim the tops and remove the yolks to present the eggs upright. You should have 12 upright eggs or 24 halved eggs.
To the hard-boiled egg yolks, add the grated onion, grating directly over the bowl to catch the onion juice, along with the garlic, relish, Worcestershire sauce, mustard, hot sauce, salt, pepper and paprika. Add the mayo, a tablespoon or two to start. Mash to very smooth, adding a bit more mayo if too dry.