Who Owns the Flag?

Whenever a suggestion is made that perhaps our enemy likes the Left's plan to retreat and surrender to them, the Left is fond of putting an an outraged air and sputtering that their dissent is patriotic and that conservatives don't "own" the American flag.

Yet when one war supporter tries to hand out small American flags to war protesters, once they figured it wasn't some post-ironic statement in support of their protest and that nobody would be following the flag distributor to pass out lighter fluid and matches, the protesters refused to touch them:

Ah, so now defeatist protests are considered "art," and passing out flags is "vandalism." Got it.

I'd be persuaded that conservatives didn't exclusively own our flag if Leftists didn't react to the flag like vampires reacting to Holy Water.