Green’s Dictionary of Slang

1. a hat.

1935		Mirror (Perth) 14 Sept. 17/2: Hat: This and that.
1944		Maurer & Baker ‘“Aus.” Rhyming Argot’ in AS XIX:3.
1960		J. Franklyn Dict. of Rhy. Sl.
1999		R. Walton ‘Cockney Jack’ 🌐 When Jack finished eating he put his this and that on his loaf of bread, picked up his Isabella, kissed his cheese and kisses, and left for work.
2002		B. Kirkpatrick Wicked Cockney Rhy. Sl.

2. (also thises and thats) spats.

1984

Partridge DSUE (8th edn) 1221: early C.20.

3. a cricket bat.

1955		Argus (Melbourne 5 Feb. 3/2: Hutton has now [...] raised his This and That, waiting for Miller to take a Currant Bun and send down the first Town Hall.
2002		B. Kirkpatrick Wicked Cockney Rhy. Sl.
2002		Pete’s Aussie Sl. Home Page 🌐 this and that: a bat, for cricket.

4. a cat.

2002

B. Kirkpatrick Wicked Cockney Rhy. Sl.