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She shook back the hood and took off down the strand in a long chain of cartwheels and flips, her coat tumbling around her shoulders.
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I can see how the juxtaposition of those phrases could be confusing. While "take off" often refers to removing clothing ("I took off my shoes and socks"), here it has a completely different meaning, namely to swiftly depart.

"shook back the hood" means that she caused the hood to fall down from her head by shaking her head/shoulders.

Comments  

She shook her head such that her hood fell off to the rear. To take off is to start running, to leave rapidly.

 GPY's reply was promoted to an answer.