My experience is that the usual arrangement of the ties is that there is a short tie on your left, and a long tie on your right. You remove the short tie from the card, and hand the card with the long tie to someone. You then turn to your left, so that the long tie goes behind your back, take the tie out of the card, and tie them on your left side.
If some ***** manufacturer made a gown with the long tie on the left, and a cheapskate hospital bought them in that manufacter's inevitable going-out-of-business sale, then it'd be reversed.
As for the card, it starts out sterile. If you hand it to someone sterile, it stays sterile, and you don't need to worry about touching anything. If you hand it to someone who's not sterile, your half is the half with the tie. The non-sterile person touches the half without the tie, and never touches the tie. After you let go of the card to spin, you do not touch the card again; just pull the tie out of it.
As for scrub techs... in the OR, they are right. Period. It's their job to maintain the sterility of the field, and they have (and should have) wide latitude to carry out that responsibility. If they tell you you contaminated yourself or something else, step away and say "Thank you." You cannot win by arguing with them. And if you cop an attitude, they will not trust you. (And you wouldn't either, in their shoes.)
On the other hand, my experience has been that there are lots of good scrubs nurses/techs who really look out for you if you're respectful and nice to them. I don't mean sucking up, I just mean treating them like professionals who know their job. And if a smart scrub nurse is looking out for you in the OR, you have a great chance to look like a star with the attending. (As Febrifuge points out, this principle applies in many areas of the hospital, but the effect is greatest in the OR, I think.)