You heard you can't have multiple types of sticks, which means that you cant have DDR3 and DDR4 on a board functioning at the same time. Even hybrid boards that are DDR2/DDR3 cant work with both types at the same time.
So what you heard is correct, you just interpreted it wrong.
Of course you can have RAM sticks of different brands and specs, but they are not guaranteed to work. RAM incompatibilities do happen, and it is recommended to use identical sticks on the same board.
However, in my experience of PC building, most of the time different RAM sticks work together just fine, even on this PC that I'm using now, I have 2x Teamgroup 1333 and 2x Corsair 1600.
Do keep in mind that if the RAM sticks are compatible, they will operate at the frequency of the lowest stick, and the latency of the higher stick, by default. For example, in my build, memory works at 1333 MHZ. All my sticks are 9-9-9, but if 1 of them were, say 10-10-10, then that is the latency they would all operate on.
One important thing: Dont even try to use different sticks on the SAME CHANNEL. Its more likely that they wont work this way.
So yeah, I'd say get the corsair ones (i've never had a compatibility issue with corsair), install them with whatever other RAM sticks you got, and see if it works.
What exactly do you have to lose? Are you looking to purchase RAM? You said you already have them, right?