Hello Jacklong

They believed that only then could Australia maintain its strength, independence and distinctive culture

This is called "dramatic inversion" or "negative inversion". You can say "Australia could only then maintain its strength, independence and distinctive culture" as a normal word order. But if you want emphasize, you would raise the adverbial phrase "only then". This operation triggers a subject-verb inversion. This kind of inversion occurs especially when the adverbial phrase connote a negative or nearly negative sense and so it is called negative inversion. Some examples are;

(1) I have never heard such a terrible lecture. -> Never have I heard such a terrible lecture.
(2) You can’t see such large forests anywhere else. -> Nowhere else is it possible to see such large forests.
(3) They didn’t realise how bad the damage was until the news came. -> Not until the news came did they realise how bad the damage was.
(4) I can only relax after I have had a drink. -> Only after I have had a drink can I relax.
(5) I was not only tired, (but) I was also hungry. -> Not only was I tired, (but) I was also hungry.
(6) Almost as soon as I got into the hotel, the tsumamis came. -> No sooner I got into the hotel than the tsunamis came.
(7) The sun rose and almost immediately it began to rain. -> Hardly had the sun risen before it began to rain.
(8) There has seldom been so much aid to the disaster's victims. Seldom has there been so much aid to the disaster's victims.
(9) He must no leave the country on any account. -> On no account must he leave the country.
(10) I wouldn’t go back to university under any circumstances. -> Under no circumstances would I go back to university.

paco