Words that break down into other valid words, even if the other valid words are not associated with the original word. For instance
banana -> ban an a, legend -> leg end, broadsword -> broads word
Unless the word breaks down into elements that are relevant to the whole, which would make it a compound word, linguists wouldn’t bother with this. Your examples are just accidents of spelling.
On the other hand, if you broke broadsword down as broad + sword, then you do have a compound word, and this would be a valid example to discuss if you were comparing the ways different languages handle compound concepts.
I was hoping there was something more general than compound word, not for a linguist purposes, but for use in questions/crosswords etc.
Eg a crossword clue like famous toe or hip (6 letter) -> legend
In cryptic crosswords, that type of clue is sometimes called a charade (after the game Charades). I haven’t heard the words themselves described as “charades,” though.
The clue like this normally contains a question mark at the end, indicating a pun.
Aren’t these just a specific type of anagram?
Sounds like a perfect excuse to invent a word. Call them wordagrams.
In Martin Gardner’s book Aha! Insight, there is a chapter with words like you are talking about. I no longer have the book in my possession so I cannot check what he called them but I am pretty sure he called them something.
Nope, nothing in Gardner’s book. There is a chapter on Word aha!, but nothing about a name for words as described by the OP. There is a discussion about the importance of an empty space to understanding a sentence or phrase (such as carson city vs. cars on city). He does refer to words that can be found within other words or phrases as “hidden words”, but that’s it.
The chapter I was thinking about had a character called “Nosmo King.” I am pretty sure I had the right book but I could be wrong. It has been a while.
Nothing to add except a few more examples:
together->to get her
tomorrow-> tom or row
mangetout->man get out
no 
mangetout -> mange tout , he sells old skin diseases in a market in Putney.
I wonder if there are any normal words with rude subdivisioning.
Subdivisioning is not a normal word. It’s a word you coined. The normal word is subdividing.