Iolanthe on her head:  a discussion

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In the opening dialogue scene of Iolanthe, we are told that the title
character has been sentenced to penal servitude for life, "And that sentence
of penal servitude she is now working out, on her head, at the bottom of that
stream!"

Question:  Is "working out a sentence on one's head" an idiom?  What does it
mean?

From: Mike Nash

I'd always assumed it meant that Iolanthe had been literally balanced on her
head, upside down. But Bradley's "Annotated G&S" states: "'On your head' was
a familiar Victorian catch-phrase meaning 'with ease'. It is still used in
the form 'You can do it standing on your head'. Gilbert's use of it here
seems rather unnecessary."

From: Bobbie

"on one's head" was a Victorian term for comparative ease.

From: Peter Zavon

I belive it means "easily" or "with ease," as in "Oh, that's so easy, I
could do it while standing on my head."

From: Philip Sternenberg

It's an idiom.  It's like "I could do that standing on my head"; i. e.
effortlessly.  Nevertheless, I wouldn't be shocked to learn that Gilbert
would have delighted in people's taking the expression to be literal,
something only a supernatural being might be forced to do for 25 years.

One THESPIS exchange uses both the figurative meaning and a slightly more
literal one.  Thespis is referring to ruling the world in the capacity of
gods and goddesses:

THESPIS:        Why, it's mere child's play!

MERCURY:        Very simple, isn't it?

THESPIS:        Simple?  Why, I could do it on my head!

MERCURY:        Ah -- I daresay you will do it on your head very soon.

THESPIS:        What do you mean by THAT, Mercury?

MERCURY:        I mean that when you've turned the world QUITE topsy-turvey
you won't know whether you're standing on your head or your heels.

(Unrelated aside:  I spelled "topsy-turvey" as I found it in ORIGINAL PLAYS,
for those of you who were arguing about the correct spelling when the movie
came out without the E in its title.  Apparently it's correct with or without
the E.)

From: Ronald Orenstein

Is there an obscure pun here?  Does the phrase "working out a sentence"
refer in any way to (say) solving a school grammar problem, so that doing
it on your head could mean solving a problem easily?

From: C. M.  Johnson

I have always assumed that "on her head" meant "on her own responsibility"
or "through her own decision".  Some folk still use the phrase "on your own
head be it" when disassociating themselves from the consequences of
another's  foolish actions.  

My reading is that Iolanthe herself selected this damp location for family
reasons.  I trust we can all agree that the poor soul wasn't physically upside
down?

From: Sandy Rovner

> I have always assumed that "on her head" meant "on her own
> responsibility" or "through her own decision". 

Yes, that is true.

> I trust we can all agree that the poor soul wasn't
> physically upside down?

No, I do not agree at all.  I certainly have always assumed that she was
indeed upside down --a grand way, after all to load the queen with a
rightside up guilt trip!

From: Diana B

The review of the first production in The Theatre definitely stated that
she was upside down and while that publication could well have misread
Gilbert's intention, it does go to show that he wasn't using a term that
would have been obvious to his audience.