Must conditional sentences begin with "if?"

I've been looking at conditional sentences (conditional clauses).

Every example I've see is along the lines of, "if [x] then [y]."

I've seen alternatives/substitutes for the if part:

• were I you (instead of "if I were you")
• on condition that
• unless
• were

Yet, these still follow the same pattern: "condition [x] then [y]."

Is that the only way to have a conditional?
Is it not possible to invert/switch around the structure?
Would it still be a conditional clause (or conditional sentence) if I put the condition after the occurance?

• If he eats that, he'll be sick.
• He'll be sick if he eats that.
• Unless you win this round, you are out.
• You are out unless you win this round.

If those are not conditionals, what are they?
Would they be considered as "acceptable" if I were to be editing/rewriting something, or would it be seen as bad/improper/incorrect/having a sufficiently different implication?

Just as Fumblefingers comments, as long as there is a conditional clause (ie a subject-verb combination beginning with if, unless, etc) in a sentence, whether it follows the main clause or precedes it, it is always a conditional sentence.

Whether you put the condition before or after is a matter of style and emphasis. Typically a sentence builds up to some sort of conclusion, so we put the thing we want to emphasize last. This is especially true if you are trying to make a point.

Compare:

If he had the courage, Bob would have asked Sally on a date.

Bob would have asked Sally on a date ... if he had the courage.

The second is more emphatic. It leaves the reader wondering for just a fraction of a second why Bob didn't ask Sally, building up a little tension. The first sentence gives the reason away immediately and so does not build any tension. This kind of subtle difference in wording and emphasis can be the difference between a simple statement of fact and an hysterically funny joke.