Actually, “the recycling” in your sentence sounds ok, Josef. It doesn't
grate on the ear. The (AmE) trash/garbage / (BrE) rubbish is
the stuff that isn't recyclable and gets buried (in landfill)
or burnt. “The recycling” means “the stuff for recycling” (tins, paper,
glass, some plastics), rather like “the washing” means “the stuff that
needs to be washed” or “the stuff that has been washed” and “the
ironing” is “the stuff that needs to be ironed” or “the stuff that has
been ironed”, then there's “the washing-up” – nádobí, etc.
Here's an example in context, where “your recycling” means “the
materials you have put out to be recycled”:
What
happens to your recycling
After your recycling is collected it is taken to the recycling facility at
Aldridge to be sorted …