Death and life lie in the hands of the tongue ...
Shlomoh HaMelech, Mishlei 18:21
מות וחיים, ביד-לשון
A traditional witch's broom is called a besom. Like besomim (sweet-smelling havdalah spices), the besom concerns "cutting".
The word besomim בשׂמים is from the shoresh בשׂם meaning "cut up" . Words are like thought-pieces cut from the stream of consciousness to communicate that contained within the inner stream to others or to oneself during the process of self-discovery. Words, like spices, can be sweet or foul smelling.
As spices are "cut" from plants like words are "cut" from thought, the besom "cuts" life from death. Like the havdalah spices specifically, the besom "separates" between realities. The havdalah besomim separate between sacred and profane realities. The besom separates between the realities of life and death. How do we know this?
In addition to the "cutting" connection of the besom with its phonetic Hebrew root, there is also a "cutting" connection within the mythology surrounding the witch's broom. In ancient Rome midwives used special brooms to "cut the ties between the world of the living and the realm of the dead after childbirth" . Interestingly, this directly calls to mind within the mythic symbolism the proverb of Shlomoh HaMelech - death and life lie in the hands of the tongue.
The besom-broom even looks like a tongue (held in the hands). Besomim are the words which fill it. When we breathe the sweet-smelling havdalah spices, we are breathing in the essence of good words born of shabbat consciousness, which hopefully will find expression throughout the week. When a witch sweeps out negative energy from her sacred space with her besom, she is "cutting off" evil words before they are even conceived.
The witch's besom in a witch's hand is the "hand of the tongue" crafting a path of life and cutting off the path of death. Besomim are the essence and expression of words the tongue may speak.
 Etymoligical Dictionary Of Biblical Hebrew, R' Matityahu Clark
 Solitary Witch, Silver Ravenwolf; and The Woman's Dictionary Of Symbols And Sacred Objects, B. Walker