Auntie running back and forth (4)

I haven't cracked this one yet — but the brilliant team at Times for the Times have:

The BEEB is a nickname for the BBC, also known as ‘Auntie’. ‘Running back and forth’ indicates the answer is a palindrome. Here’s how ‘Auntie’ came about according to one source:  In the early years of BBC development the administrators were largely of middle class origins and demonstrated daily that they knew what was best for listeners and later on the viewers. This was derided by critics as the “Auntie knows best” syndrome. Hence the shortened version of “Auntie”. My own recollection is that it didn’t start until the 1960s but that may just be when I first became aware of it.