O do reverse at an order of the dragonfly! (7)

I believe the answer is:

odonata

I'm a little stuck... Click here to teach me more about this clue!

'order of the dragonfly' is the definition.
'Odonata' can be an answer for 'order' (Odonata is an example). I am not certain of the 'of the dragonfly' bit.

'o do reverse at an' is the wordplay.
'reverse' says the letters should be written in reverse.
'at'+'an'='atan'
'atan' back-to-front is 'nata'.
'o'+'do'+'nata'='ODONATA'

Can you help me to learn more?

I've seen this clue in the Sydney Morning Herald.

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