I have always said the following should be paired together:
1...e5 - Nimzo-Indian as both are strategically rich and lead to diverse pawn structures
Caro-Kann - Orthodox QGD - both very solid and a safe king take priority over attacking White.
French - Dutch/KID - The Dutch via 1...e6 or the Kings Indian with it's blocked nature both pair up well with the French.
d6-Sicilians - Grunfeld - Highly theoretical and Black playing for a win from move 1
Nc6-Sicilians - Slav - Both solid with a tinge of aggression
e6-Sicilians - QGA - Both seek free piece play with little to no static pawns.
As the title suggests... we usually don't talk about which combination of 1. e4 and 1. d4 defenses should be played together. In particular, paired in a way that will best handle the English / Reti. This turns out to be pretty significant when choosing the repertoire and worth thinking about. It's not something I've thought about much but here goes... maybe some other people also will have something to say on this topic.
If you're a QGA player - you can rarely reach a QGA from the Reti or English. So you need some unique lines in the Reti / English, you don't really want to play another d4 defense. And a good candidate here is the Kings English, a reversed sicilian. Another good candidate in the Reti is the Sicilian Invitation (1... c5). But to do this you ought to play the Sicilian, especially for 1... c5 in the Reti which can just transpose right into the open sicilian after 2. e4. Therefor I think we could conclude the QGA and Sicilian pair well together in a repertoire. Generally any sort of 1. d4 defense that won't work in the English or Reti such as the Albin or the budapest, also will pair well with the Sicilian.
If you're a french player... against 1. d4 you can respond with 1... e6, and if 2. e4 you're in the french. But if 2. c4 you can respond with b6 and play an english defense, this is a very dynamic opening and it scores well especially at club level. In turn the English opening pairs well with the QID, which can be played against the Zuckertort. Then against the English you've got your 1. b3 opening, which throws a wrench in whites plans to play g3. And against the Reti you have a variety of things you can transpose into, including the QID. Another option after 1... e6 is to play a delayed dutch.
Okay, so I'd say QID / french / English defense / Dutch are another set of openings that pair well together. And perhaps if you want something more solid you can also add the nimzo indian here as well.
Now... for the caro kann... I think the slav / semi-slav is pretty obvious and doesnt need alot of explanation. Though I think you could get away with pairing the semi-slav with something else against e4, you may just need a few caro-kann exchange / panov lines, but this should still be fine.
So what about e4/e5 then? There's not alot which transposes into e4/e5. So you need 1. d4 defense that can handle the English / Reti on their own. Which means an e4/e5 player should not play the QGA, or any of the attacking sidelines against 1. d4 such as the albin or budapest - because you won't be able to use those against the English / Reti and you'll be left with nothing.
Agree? How do you guys handle the English / Reti in your repertoires?