Is there an integer that has its digits reversed after dividing it by 2?

Assume there's such a positive integer x such that x/2=y and y is the
reverse of x.

Then x=2y.  Let x = a...b, then y = b...a, and:

                 b...a   (y)
               x     2
               --------
                 a...b   (x)

From the last digit b of x, we have b = 2a (mod 10), the possible
values for b are 2, 4, 6, 8 and hence possible values for (a, b) are
(1,2), (6,2), (2,4), (7,4), (3,6), (8,6), (4,8), (9,8).

From the first digit a of x, we have a = 2b or a = 2b+1.  None of the
above pairs satisfy this condition.  A contradiction.

Hence there's no such integer.