The short answer is yes.
If the question was; do we recycle deodorant tubes? The answer would be no.
Few plastic products that contain a variety of different plastics are currently recycled. This is due to the fact that plastics of different types or grades do not mix together when melted down. (The average deodorant tube contains four different type of plastic). This means that each tube would have to be separated into its individual types of plastic before it could be recycled.
It is currently easier and cheaper to throw them away and make new ones from raw polymers that are created from oil.