The most common and most likely answer to this crossword clue is the 6 letter word PLEACH
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More clues leading to the result PLEACH
Buddy explains!

The straight part
We know that "Intertwine branches" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case intertwine branches = pleach.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does PLEACH mean?
PLEACH noun- An act or result of interweaving; specifically, a hedge or lattice created by interweaving the branches of shrubs, trees, etc.
- A branch of a shrub, tree, etc., used for pleaching; a pleacher.
- A notch cut into a branch so that it can be bent when pleaching is carried out.
- To unite by interweaving, as branches of shrubs, trees, etc., to create a hedge; to interlock, to plash.