Sort rows by the weight of a bouncing diagonal path

In a square matrix, start a path at each cell of the first column. The path steps one column right and one row up each time; once it reaches the top row it turns and steps one column right and one row down until it reaches the last column. A start cell's weight is the sum of the values on its path. Sort the start cells by weight, breaking ties by the start cell's own value, and return the start values in that order.