The term 'Dismisses Unceremoniously' can be broken down into two crossword solutions: 'BOOTS' (5 letters) and 'AXES' (4 letters). 'BOOTS' is a term used to describe the act of firing someone from their job, while 'AXES' is a term used to describe the act of cutting or eliminating something. Together, these two terms can be used to describe the act of firing someone from their job without any ceremony or formalities.