I know nothing about Australian politics.
Yes, you do. I assume you're at least aware that it's a parliamentary constitutional monarchy with norms resembling those on other advanced western democracies.
Am I allowed to refer to myself as non-ideological when it comes to Australian politics?
You have an ideology about how governments ought to work, you may not know enough about Australia to know what its doing wrong or which party is closest to pushing things in the wright direction, but if you did the research you'd either find yourself sympathizing with one of them or conclude that all of them are insane and advocate some radical (from the point of view of internal Australian politics) position that hardly anyone there is advocating.
Let's replace Australia with something like Saudi Arabia, North Korea, Belarus, or Kazakhstan. I'm guessing you probably know about as little about at least one of those countries as you know about Australia and yet you probably have opinions about what's wrong with those countries.
To take my point even further: do I have an ideology regarding the best way to use a bucket full of Legos?
You have priors and preconceptions about it.
What is it about politics that causes everyone to have an ideology a priori?
Because the word "ideology" is generally used to refer specifically to politics.
Politics as gymnastics for rationalists. No one one Less Wrong is quite sure why politics is a taboo topic or how things got to be that way. What we do think we know is that politics is a great way to bring out the irrationality in people. So why not take advantage of that and use politics as a way to measure rationality? Since politics brings out the most irrationality, it should provide the strongest signal. Since there aren't useful objective metrics of how a political discussion went, we'd have to use subjective judgements by neutral third-party raters, kind of like they do in gymnastics. (In the comment thread for this post, feel free to find fights that you have no dog in, improvise a rationality rubric, and grade participants according to your rubric... let's see how it goes.)
Be a sheep. This is probably the exact opposite of what you were taught in your high school civics class. But if my friend Jane is more intelligent, more informed, and less ideological than I am, it seems like voting however Jane is going to vote is a strict improvement over voting however I would naively. It also saves me time, and gives Jane an incentive to put even more time in to carefully considering political issues since she now controls two votes instead of one. Done on a large scale, this could provide an interesting twist on representative democracy. Imagine a directed graph where each node represents a person and an edge is directed from person A to person B if person A is auto-copying person B's votes. There's a government computer system where you can change the person you're auto-copying votes from at any time or override an auto-copied vote with your own personal guess about what's best for society. Other than that, it's direct democracy... all bills are put before all citizens to vote on. Problems this might solve: