Palindromic numbers with a binary twist
05AB1E, 7 bytes

b0Ü‚DíQ


Try it online!
or as a Test Suite

Explanation

b         # convert input to binary
 0Ü       # remove trailing zeroes
   ‚      # pair with input
    D     # duplicate
     í    # reverse each (in the copy)
      Q   # check for equality