A few words in The World of Twirls:

Curl (Herman van Goubergen has used this term in Curler Unit article)
If you curl a flap it is curved or spiral in shape.
If a person or animal curls into a ball, they move into a position in which their body makes a rounded shape.
When a leaf, a piece of paper, or another flat object curls, its edges bend towards the centre.
The corners of the lino were curling up.

Twirl
If you twirl something or if it twirls, it turns around and around with a smooth, fairly fast movement.
If you twirl something such as your hair or paper, you twist it around your finger.   

Twist  
If you twist something, you turn it to make a spiral shape, for example by turning the two ends of it in opposite directions.
A twist is the shape that something has when it has been twisted.
 

Curve
A curve is a smooth, gradually bending line, for example part of the edge of a circle.
If something curves, or if someone or something curves it, it has the shape of a curve.
If something curves, it moves in a curve, for example