Word pairs that describe something in both orders.

Back in the day, American fencers racked up Olympic medals in singlestick – but these days you’d need a fencing sword, because it’s now limited to sword fencing.

Head butt and butthead.

I’d like to write a horror novel, but I can’t think of a novel horror.

You may be better at knife throwing, but I’m the one with the throwing knife.

How about baby fat and fat baby?

A woman knows she’s getting old when people stop telling her “You’re good looking” and start telling her “You’re looking good”.

There’s even an old song about that.
“I’m looking over
A four-leaf clover
That I overlooked before . . .”

Yes, I think the spirit of the OP is that both orders of the words should be recognised phrases rather than just two words that can be used in that order in a sentence.

So if it’s specialized jargon it doesn’t count?

ETA: I think out punch could fairly have been spelled “out-punch,” and I think it counts. If people want to say “top lap” doesn’t because only geologists or archeologists would recognize it, I’ll accept that.

ETA2: I think “chicken fat” and “fat chicken” count, but that may be just me. I’ll wait for the OP to rule.

glass eye, eyeglass
cake pan, pancake
stepladder, ladder step
side ring, ringside
barrow wheel, wheelbarrow

You look seriously over worked.

Johnny really got **worked over **by those Corleone goons.

I heard he wound up needing a donor organ from an organ donor.

They say he had to hock the watch crystal from his crystal watch.

Would you rather be:

Up beat or beat up?

Off beat or beat off?

I’m tempted to say “Kitty Genovese,” and “Genovese kitty,” as in, a cat from Genoa, but I think that would be very wrong.

boot camp and camp boot (i.e. a boot you might wear when going camping)

weed killer and killer weed

hopscotch and Scotch hop

Snow White and white snow

A lot of other color words will get you this:

red firetruck and firetruck red
red beet and beet red
gray slate and slate gray
blue sky and sky blue

A test pilot – “jet fighter” or “fighter jet”, it makes no difference – ain’t a pilot test.

I left my security job for a position with more job security. Which is to say: I earned a degree in museum science, and got hired at the science museum.

We have a lot of examples like that here in northeastern Kansas – the KC-based Hall family (of Hallmark fame) spread their wealth generously.