[See the Note below: “2010 data re-posted.”]

The above graphs plot the time series of monthly data for the whole of Australia, smoothed with a Gaussian window of half-width 6 months.
The two independent series of (a) mean maximum monthly temperature anomaly in degrees celsius and (b) total monthly rainfall anomaly in mm are plotted on the same graphs, but the scale for temperature is inverted for easy comparison.
One degree on the left axis corresponds to 10 mm on the right axis, but the zero lines may differ.

There is an obvious sine-wave cycle with a wavelength of between one and three years. (This is the “quasi-biennial cycle” that A.B. Pittock identified in 1971.)
Most peaks and troughs on these independent time series almost coincide, and their relative heights and depths tend to agree. In fact. the correlation between values of temperature and rainfall is poor, but the shapes of the sinusoidal curves match extremely well.
Peaks and troughs on the rainfall curve tend to lead those on the (inverted) temperature curve by one, two, or three months. In these graphs, I have lagged rainfall values one month, to show that many of the peaks and troughs are aligned.

Taking the whole of Australia in the last 60 years, it is fairly clear that:
* points of lowest maximum temperature have generally lagged about one month behind points of highest rainfall;
* points of highest maximum temperature have generally lagged about one month behind points of lowest rainfall.

Given this lag effect, times of lowest rainfall cannot be caused by times of highest temperature, but it is possible that times of highest temperature may be caused by times of lowest rainfall.
I find it plausible that temperature swings would closely follow rainfall swings (but in the opposite sense) due to lack of cooling by evapotranspiration in times of drought and effective cooling by evapotranspiration in times of deluge.

[Note: 2010 data re-posted.
This material appeared originally in a “Weatherzone” forum “Observations of climate variation”, Post #810237 of 27 December 2009.
Since the graphs as posted are lost, due to the action of the “Photobucket” image-hosting web-site, I am re-posting the graphs from my records.]

Data source.
The data was sourced at the following web-page.
http://reg.bom.gov.au/silo/products/cli_chg/
That web-page no longer exists.

[Note posted to “weatherzone”.

I never did find Barrie Pittock’s 1971 article in which he referred (I believe) to a quasi-biennial oscillation in Australian surface climate.
However, here is a detailed discussion of that particular topic:

“Historical El Nino/Southern Oscillation variability in the Australasian region” by Neville Nicholls, Chapter 7 (p151-173) in “El Nino: Historical and Paleoclimatic Aspects of the Southern Oscillation”, Henry F.Diaz and Vera Markgraf (eds.), Cambridge U P, 1992, 476pp.

On p.158, Nicholls has a section headed “Biennial cycle” that refers to papers written in the 1970’s, 1980’s, and 1990’s. He says:

“The biennial cycle is observed over the equatorial Pacific and Indian Oceans and is tightly phase-locked with the annual cycle. It varies in amplitude from cycle to cycle and sometimes changes phase. It is not strictly a 2-year cycle so it may be characterised better as a quasi-biennial cycle…..Rainfall over much of Australia displays a quasi-biennial cycle (e.g.Kidson 1925).”]