Use the Left-Hand Rule. see LHR#2 on this Wiki page:
http://schools.wikia.com/wiki/Left_hand_rules
Use your LEFT and and wrap it around the winding you are making so that your finger tips point in the direction of CURRENT flow. Remember that the convention for Current is that it flows from positive to negative, not the reality of electron flow. Now the thumb of you left hand fist point to the NORTH pole of the electromagnet you've made.
To your original question, then suppose you fold the wire in half and wrap it around a core. Now imagine For the first half of the coil, the current flows in one direction. But at the end of the coil it suddenly reversed around the fold and stars flowing in the opposite direction in space. So the fields produced by the two halves of your coil cancel each other.
If you cut the wire into two pieces and wrap them in parallel around your core at the same time, you basically create two windings in parallel around a single core. Then what? Well, if you connect wire ends together so that they are operating in parallel electrically, then the two coils' fields reinforce each other and you still have an electromagnet. If you run an alternating current through ONE coil only and then connect a load across the ends of the other, you'll find current flowing through the load. You have just built a transformer!