Here’s my take. Just as a bunch of ideas, perhaps it’s helpful.

Diaphonemes(F)	Phonemes(F)	Diaphonemes(B)	Phonemes(B)
æ(trap), ɑː/æ(bath)	/æ/	ɑː(palm), ɒ(lot)	/ɒ/
ɜ:r(nurse)	/ær/	ɑ:r(start)	/ɑr/
ɛ(dress)	/e/	ɔː(thought), ɒ/ɔː(cloth)	/o/
eɪ(face)	/ej/	oʊ(goat)	/ow/
N/A	/ew/	ɔɪ(choice)	/oj/
ɛər(square)	/er/	ɔ:r(north)	/or/
ɪ(kit), i(happy)	/ɪ/	ʊ(foot)	/ʊ/
ɪər(near)	/ɪr/	ʊər(cure)	/ʊr/
i:(fleece)	/ɪj/	u:(goose)	/ʊw/
ʌ(strut), ə(comma)	/ɐ/	ʌ(strut), ə(comma)	/ɐ/
aɪ(price)	/ɐj/	aʊ(mouth)	/ɐw/
ər(letter)	/ɐr/	ər(letter)	/ɐr/

All sets were reorganised into a “pseudo-dialect” with seven vowels, no length. It shouldn’t correspond to any RL dialect, but still be close enough to be English. Almost all phonemes were paired, but:

• /oj/ got an artificial front counterpart, /ew/.
• when inverting any /Vr/ sequence, sub the /r/ with the consonant you choose as /r/'s “opposite”.
• /ɐ/ is its own opposite. It’s a “neutral” vowel.

I thought of that… I just think it would get too messy if I tried to be precise with this method. Maybe I just need another method.

Personally I’d go with the merry-go-round. Because it’s somewhat easy to force English into three sets of consonants, through the following changes /θ ð h/ → /f v Ø/ (all three associated with baby talk). Then you get the following:

labial: /m p b f v/
apical: /n t d s z/
dorsal: /ŋ k g ʃ ʒ/

The only ones left behind are /t͡ʃ d͡ʒ l r j w/. I think /j w/ can be paired together (as if they were vowels), /l r/ (they often occur in similar environments), and then /t͡ʃ d͡ʒ/ can be analysed as sequences, so after “opposed” they end as /ps bz/, /kf gv/ or something like this, you get the idea.