If a hermaphrodite had sex with itslef, would it be able to have both a male and a female orgasm?
Sure.
Was that all you wanted? 
(S)he would scream “GRY!” while climaxing, and it wouldn’t echo.
Seriously, I think the whole idea that male and female orgasms are fundamentally different went out the window with Freud.* Orgasms are orgasms, and since hermaphrodites have physical characteristics from both sexes (they don’t have two sets of sex organs, as you may have assumed), I’m sure that they have orgasms as unique as they are, just like you.
*[sub]Look out below![/sub]
Fundamental primordial tissue differentiates by sex. With very very rare exceptions, no hermaphrodite has multiple copies of any of the primordial tissue with one copy differentiated into female and the other copy differentiated into male.
In other words, you might have a scrotum or you might have labia, or you might have something that’s sort of in-between (a ball-sac with descended gonads and with some degree of separation/performation along the seam), but you’d be extremely unlikely to have a scrotum and also, separately, labia.
You might have a penis, or a clitoris, or something midway in between, which could be free-hanging from the base or attached and embedded in surrounding tissue or perhaps detached from the hood/glans on down quite a ways but not all the way down to the base as with a penis; the base might be bifurcated like a wishbone, or not. But you’re most likely going to be allocated no more than one of them, regardless of its configuration. You would not likely have a penis and also a clitoris.
It is remotely possible that you could have a testicle on one side and an ovary on the other, since generally people have two such items, but this would also be highly unusual.
Anyway, orgasms: that would be predominantly from stimulation (direct or otherwise) of the glans/hood tissue with all the wealth of nerves. And you’d almost for certain have only one such place, whether that place were mostly akin to the hood of a clitoris, mostly resembled the glans of a penis, or looked like a cross between the two.