Does anyone know if I can stack gladiator oaths and siphon of anguish or will only one of the set of affixes apply?

I am not sure I understand the question because they don’t compete for the same item slot.
You can equip both and benefit from both.
You can have as many uniques as you want

Some stats stack (like attack speed & damage), some don’t. Though all of the stats from both those uniques would stack (with the likely exception of the Arena creation on Dancing Strikes 3rd strike).

@PvtVlad I think he just wants to know what would happen if he equipped two of either of those uniques.

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The question is because the name is “unique” and some games only allow you to equip/use only 1 of any given unique at a time.

LE doesn’t do that. You can equip multiple of the same unique and get the benefits from both.