The most common and most likely answer to this crossword clue is the 12 letter word BRILLIANTINE
I've seen this crossword clue in Financial Times
B
R
I
L
L
I
A
N
T
I
N
E
More clues leading to the result BRILLIANTINE
Buddy explains!

The straight part
We know that "hairdressing" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case hairdressing = brilliantine.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 BRILLIANTINE mean?
BRILLIANTINE noun- A hair pomade, making the hair shine brilliantly.
- A smooth shiny, luxurious fabric, often of alpaca or vicuña.
- To apply brilliantine to the hair.