Watch this girl trampolinist at the Rio Olympics from about
3:00 in the video, and I'll come to your swing example after that
See how she rotates faster when her legs and/or arms are tucked in and slower when extended out. This doesn't happen because she throws her legs and arms
"forwards", it happens because she is extending her limbs
"outwards",
away from her centre of rotation to slow down and inwards,
closer to her centre of rotation to rotate faster.
What is happening is that she is repositioning some of her mass. That mass has a certain momentum (called "angular momentum" because it is not in a straight line). It maintains that momentum when you move it closer or further away from the centre of rotation. When you move it closer it still has the same momentum, but now it is covering the same distance on a much smaller circle than it was previously - the momentum cannot increase, so the rotation rate increases to conserve the momentum - when the mass is moved outwards, again, the momentum cannot decrease, so the mass has to travel a larger circle, and the rotation rate slows. This called conservation of angular momentum
I can only have one video per post, so I'll answer the swing question in the next post...