The primary meaning of the Australian-English phrase not to know whether one is Arthur or Martha, and of its variants, is to be in a state of confusion.
These are, in chronological order, the earliest occurrences of this phrase that I have found:
1-: From Bar One: Name to be Changed, published in The Inverell Times (Inverell, New South Wales) of Friday 16th May 1941—as deciphered from the original document, which is almost illegible:
—Context: Another name was to be chosen for the racehorse Bar One, because this name was deemed likely to cause confusion among the public and bookmakers:
Bar One is engaged at Gosford on Tuesday, and should, by any chance, this horse figure prominently in market operations, backers and bookies won’t know if they are on Arthur or Martha.
2-: From Euchre, by ‘Alone’, published in The Grenfell Record and Bland and Lachlan Districts Advertiser (Grenfell, New South Wales) of Thursday 21st February 1946:
A large number of euchre players gathered at the Boomerang Club on Monday night for the first night of the aggregate.
We thought Mrs. Clifford was having a good night, as she visited every table, then someone told me it was only to collect the money. Well, excuse my mistake, as I was so weakened by superior play that I did not know if the name was Arthur or Martha.
3-: From the caption to the following photograph, published in Truth (Sydney, New South Wales) of Sunday 14th March 1948:
FUN GALORE IN TRIALS
The late lamented Mr. Rafferty would have jumped with joy had he been at Erskineville Oval yesterday, when Eastern Suburbs and Newtown staged trials for the coming League season. Players were all over the place like Brown’s cows 1, and most didn’t know whether they were Arthur or Martha. Still, the season’s only beginning, and Rafferty will be put to shame later.
1 About “all over the place like Brown’s cows”: the following definition is from A Glossary of Words in use in the Counties of Antrim and Down (London: Published for the English Dialect Society by Trübner & Co., 1880), by William Hugh Patterson (1835-1918):
All together like Brown’s cows, or Like Brown’s cows all in a lump, a comparison.
4-: From a letter to the Editor, published in the Border Morning Mail (Albury, New South Wales) of Tuesday 1st June 1948:
Sir,—The statement by Cr Schneider in your report of the last meeting of Hume Shire Council that Lavington Progress Association had been “kept up to date with information” about the water supply scheme is balderdash. […]
The position is that residents of Lavington do not know whether they are Arthur or Martha. They do not know by what amount the original estimate of cost of the water supply scheme will be increased, they do not know if it would be cheaper to get the water from Albury City Council or if this plan was given full and mature consideration; whether ample provision is being made for continual expansion of the area to be reticulated; if it is possible to secure larger inlet pipes to properties than the rumored three-quarter inch limit, to permit of market gardening by irrigation; and many other questions of definite interest to them.
The phrase not to know whether one is Arthur or Martha, and its variants, have come to be also used with reference to sexual orientation or gender identity. These are two examples of this use:
1-: From They Hosed Them Out (Sydney: Australasian Book Society, 1965), by ‘John Beede’, pen name of the Australian author John Bede Cusack (1908-1979):
As we passed out over the English coast […], I looked down and saw an amazing sight; a real surf was rolling onto a white sandy beach guarded by towering headlands. ‘Cripes,’ I cried, ‘there’s a surf down there. What’s that place, Jack?’
After a pause he said, ‘Newquay.’
‘That’s the place I’m going to.’ There were real howlers amongst those waves.
‘Probably freeze your knackers off,’ said Bill sceptically. ‘I only tried swimming once over here and didn’t know if I was Arthur or Martha when I came out.’
2-: From the caption to the following cartoon illustrating Not For Lesbians Only, “an extract of a speech given to feminists at a Socialist-Feminist Conference in Ohio in July 1975, about ideas on lesbian-feminist politics”, published in Camp Ink 2 (Sydney, New South Wales) of December 1976:
not caring whether you’re Arthur or Martha
2 Camp Ink was the magazine of the Campaign Against Moral Persecution (a.k.a. CAMP), Australia’s first national gay rights organisation.