They aren't quite the same.
"As opposed to" usually suggests something is an alternative, for example:
The box is made of plastic, as opposed to wood.
This suggests that some boxes are made of wood, but this box is made of plastic instead. It doesn't in any way deny that other boxes are made of wood.
"Contrary to" usually suggests an opposite, but also that whatever it is being compared to may be false, for example:
Contrary to opinion, our boxes are made from plastic, not wood.
This means that the boxes are made from plastic, not wood - but also refutes any idea that they were made of wood.
In answer to your question then - you can open a sentence with "contrary to[x]", because you will immediately go on to state both the incorrect notion and the correct one.
It is not common to open a sentence with "as opposed to". "As" is here being used as an adverb, and while you can open a sentence with an adverb, it is much more common to begin a sentence with the conjunction "as", for example "As it is raining today, we will stay inside". Opening a sentence with "as opposed to" feels wrong to me as an English speaker because of that ambiguity. It isn't clear.