It's a two-stage process.
The first is breaking down the hair's structure with a waving lotion containing reducing agent (usually thioglycollic acid), which is almost as dangerous in its own right. The second is fixing it in place with a neutraliser containing an oxidising agent (usually, as you say, hydrogen peroxide).

Chemical straighteners ("relaxers") use caustic soda - which is more typically used to unblock drains or shift really stubborn grease.
( The Mock TurtIe ™ --- Thinks you are a cunt, on, Tue 23 Oct 2012, 20:57, 1 reply, 13 years ago)

I'm going back a bit here
But wouldn't thioglycollic acid in thermite make a basic explosive?

If I'd put my granny head down in an improvised tube I could probably have set a trajectory and fired her at Morecambe.
( The Disappointed Caravanner of the year, Tue 23 Oct 2012, 21:06, Reply)

No, no,

Head down on Malcolm, not Morecombe. He does the aiming. Cos yer granny can't see where her erse is aiming.
(Phudbree oh man, the Doric Yorick, Tue 23 Oct 2012, 21:13, Reply)

Probably could, it's a strong reducing agent.
But the thioglycollic acid would have been inactivated when they put the neutraliser on, so your granny's hair was unlikely to have made a good explosive - unless they made a total hash job of it.
( The Mock TurtIe ™ --- Thinks you are a cunt, on, Tue 23 Oct 2012, 21:15, Reply)

Get the Hydrogen Peroxide in then
The Eastern Block used Hydrogen Peroxide routinely as missile propellant and even submarine propellant.

I reckon I could have got her to Burnley with that.

Thermite and thiogloccic makes a loud but localised bang - railway detonators.
( The Disappointed Caravanner of the year, Tue 23 Oct 2012, 21:23, Reply)

HTP or High Test Peroxide is still used in rockets
Not in torpedos any more. Many years ago I took the safe handling course for HTP - it's very nasty stuff - over 90% H2O2! The peroxide in hair products is extremely tame in comparison, about 3-5%.
(Captain Placid 24/7 ball gags, brownie mix and clown porn, Tue 23 Oct 2012, 22:28, Reply)