Is there something strange about seeing a great comedian who identifies as a trans woman do a solo Hamlet, the great mirror play for an actor? Eddie Izzard plays male roles when she chooses, as if gender were a glove for fitting or discarding as the role dictates. Her Hamlet is at once bizarre and dazzling. Yes, Sarah Bernhardt famously played Hamlet. Dame Judith Anderson, revered as Hitchcock’s Mrs Danvers in Rebecca and as Euripides’ Medea, mighty too as Lady Macbeth, did a solo Hamlet in 1970 at the age of 73. Anthony Hopkins, who was playing the King to Nicol Williamson’s Hamlet, never had a shot at Hamlet, though his fellow Welshman Richard Burton held the record for the number of performances for his run of Hamlet on Broadway in 1964.

Eddie Izzard devours the role of the Prince with a soaring technique which brings to vibrant life the thousand fragments of the shattered mirror she makes of this invitation to prodigious self-portraiture. The effect is remarkable. Izzard has a bewitching glamour in this play about a self in search of a signature.

From the opening apparition, she has the authority of a magic maker who is bent on reconfiguring a masterpiece. The initial gaze towards the audience is at once insolent and ripe with authority. This is the face of a Hamlet to be reckoned with – not least because she intends to wrestle the historical and histrionic crown to the ground.

The voice sounds a bit Midlands; the show can at first seem a bit throwaway, a bit self-conscious, and then gradually it creates a world of contentious occasions. It’s almost as if the ghosts of a thousand Hamlets were fluttering to life. The effect defies conventions. Izzard lords it over the script she steals and makes her own.

Sometimes Izzard modifies it – she says, as Shakespeare doesn’t, that the ghost comes from purgatory and that when things are north-northwest she knows a hawk from a heron. Shakespeare says hawk from a handsaw or heronshaw (the emendation is clearer but less rich).

But who cares if this is a deconstructed Hamlet when it is such a rich interpretation. Simon Gray, who wrote Butley, said, ‘Hamlet is one of the greatest master works of the human spirit etc the completest expression of Western man’s consciousness… which speech by speech it possibly is… The truth is that Shakespeare, although indisputably the greatest genius of moment-to-moment drama etc, and the greatest poetic intelligence in the history of the world…’ – Gray goes on to say what’s wrong with the Bard (those ghosts and witches from the sources).

Izzard turns her back on the haunting tradition of Hamlet, but the effect is transformational, and it highlights, with an extraordinary iconoclasm, what this story would want to tell if it were retold with the gifts of a great comic intelligence. There’s Polonius offering a rationale for Hamlet’s behaviour, ‘Your noble son is MAD’, and the prose clarity of the statement cuts the night air. Izzard peoples the world of Hamlet on the wings of her technique. ‘Country matters’, the ribald lunacy to Ophelia, is done with an unforgettable grating emphasis and this self-spun production – a bit amazingly – gives Ophelia her due. Not only does the Valentine scatology (‘By cock they are to blame’) have its own brilliance but Izzard actually brings alive the all-but-impossible elegiac lyricism of the Queen’s homage, ‘There is a willow grows aslant a brook.’

It is, in fact, amazing what she brings to life. George Bernard Shaw had a famous dispute with the actor John Barrymore in which the creator of Henry Higgins and Eliza Doolittle said, ‘You even cut the recorders.’ Well, Izzard doesn’t. She taunts Guildenstern and concludes, ‘Do you think I am easier to be played on than a pipe?’

Nor does she eliminate Hamlet’s last soliloquy – ‘How all occasions do inform against me’ – as Gary Taylor did in the Oxford Shakespeare. It’s remarkable how much text Izzard gets into her revamped Hamlet, and if the revamp is weird it can also have a clairvoyant quality because the words are piercingly clear as she cries for Hecuba, or says the words must be done trippingly, or says right at the end, ‘The point envenomed too!’ in that extraordinary duel which she performs all alone. Mallarmé’s description of it as a ‘sumptuous and stagnant exaggeration of murder’ is not less telling for Izzard duelling – pyrotechnically – with herself.

Yes, it’s a transposition. So are Richard Tognetti’s adaptations for the violin. But Janis Joplin’s ‘Summertime’ is not inferior to Leontyne Price’s operatic version.

Look, I know Hamlet. I listened to the studio recording of Burton’s 1964 Hamlet when it was played on commercial radio. I know why Time magazine would say – also in 1964 – that Burton pleased the ear but did not wring the heart the way Paul Scofield does, and I also know why Peter Porter, that great Australian poet, was so underwhelmed by Scofield. The Hamlet I would most like to have seen is Peter O’Toole’s, which opened the National Theatre in 1963.

The one that reached the largest possible audience – kids could get it on swap cards – was Olivier’s 1948 film (which is also introverted and contrary to the actor’s nature). In any case, it led the great Russian Hamlet, Innokenty Smoktunovsky, to say to me in English, in Melbourne, jumping ahead of his interpreter, ‘I wanted to be better than Laurence Olivier.’

Izzard’s Hamlet has the audience on the edge of its seat. Her movement is extraordinary, and the script – brilliantly edited – is rather fine. It springs palpably from a great actor’s desire to shape dynamically in the idiom of the spectrum of her art.

There will always be preferences with Shakespeare. You might prefer the English actor Stephen Dillane doing his one-man show of Macbeth (which we saw in Adelaide in 2006). It had the authority which came from his command of verse. But these things change. Peter Hall once said that Richard Burton and Gérard Philipe, who transformed the performance style of Shakespeare, would appear mannered now.

All of this feeds into what we make of Izzard, but it should not stop us from recognising what she has done: a magnificent and pyrotechnical achievement which will have the audience screaming with laughter and dumbstruck at the images one human being can allow to take possession of her extraordinary versatility.