The most common and most likely answer to this crossword clue is the 7 plus 5 letter word CRYSTAL BALLS
I've seen this crossword clue in The Daily Telegraph
C
R
Y
S
T
A
L
B
A
L
L
S
More clues leading to the result CRYSTAL BALLS
Buddy explains!
The straight part
We know that "The spheres of the occult" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case the spheres of the occult = crystal balls.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.