Splitting a clock face into three equal-sum parts

A clock face carrying the numbers $1$ to $12$ falls and breaks into three pieces. The numbers on each piece add up to the same total. Which numbers lie on each piece?

$\{11,12,1,2\}$, $\{3,4,9,10\}$ and $\{5,6,7,8\}$. The numbers sum to $78$, so each piece totals $26$. Pairs facing each other across a horizontal line add to $13$ ($12+1$, $11+2$, $10+3$, …), so two parallel horizontal breaks give three bands of two such pairs each.