Teaser 2837: Back and forth

From The Sunday Times, 5th February 2017 [link]

George and Martha have a digital 24-hour clock that always displays the time as four digits (e.g. 2:17am displays as 0217). During one lazy weekend George noted two particular times when the four-digit display was palindromic. He calculated the number of minutes from the first of these times to the second and he discovered that the answer was a three-figure palindrome. When he reported all the details to Martha, she commented that the sum of the eleven digits in the three palindromes was also palindromic.

What were the two palindromic times?