Cutting back my cypress hedge... hard

[Major T]

I've got a 7ft high cypress hedge in my front garden that hasn't been properly trimmed over the last few years (by the previous owners)
It's developed a bulging, bloated look, and I want to cut it back so it's straight and flat. This will mean cutting it back by 30-60cm.
I'm expecting it to look sorry, bare and brown afterwards, but will it regrow?
Tried googling to no avail. Any advice? Perhaps a phased approach is best? I'd like to keep it if poss.

Cheers

[Mr GrimNasty]

Most hedging conifers are happiest if you don't trim past the green, regular light maintenance trimming is best. Some may regenerate from bare wood (Thuja) especially when young but I wouldn't count on it, but Leylandii definitely won't ever.