Sometime last year, while sitting at my desk, my spouse took a photograph of me. You could figure the date and time the photo was taken because sitting beside me was my digital clock displaying the current time and date (at the time) in the format [hh:mm:ss | MM/DD].
I noticed later that the time and date displayed was the mid-point between the earliest date-time that could be displayed on my clock using only the digits 0-9 once each and the latest date-time with the same constraint.

In the photograph, what time did my clock display?

(Assume, for the problem, that the year in question is neither a leap year nor does the region observe Daylight Savings.)