The dancer rose on tiptoe and spun round and round.

Literal

Dancer [topic-は] toe-stand-[and-て] spinning [adverbial-と] turned.

つま先立つ ('to stand on tiptoe') — つま先 ('tiptoe') + 立つ ('to stand'). ぐるぐる is a 擬態語 (state mimetic) for continuous rotation — used for spinning, dizziness, winding. Very common in manga/narrative.