Just a reminder that palms are blessed and therefore sacramentals. Throwing them away would be sacrilege. Best thing to do is either bury or burn them when you’re done with them. You can also give them back to your parish
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Our church takes them back and reuses them in an anointing oil.
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I was just on vacation and I saw a Palm branch on the ground and I thought to myself, Someone did not dispose of this properly and then I realized I was in the Bahamas and they had recently had strong winds blow through 
My church collects old palms and burns them for use on Ash Wednesday.
Ours does as well. This is a fun fact for those who might not be Catholic. Thank you for pointing this out : )