Slicing up line features with a grid

I have a contour feature class. Many of the contours are single lines which snake around the entire extent making very large features (with very many vertices). I want to experiment with performance by slicing the contours up with a grid of squares. I have a grid feature class with 15x30 squares. How do I split up the contour line features without including the grid in the resultant feature class? A Union would add the grid into the result. A Clip would only clip along the outer boundary of the grid class, not the internal lines of the grid. It seems simple, but I'm not seeing an obvious tool.

I suppose I could make a script to one-by-one select squares of the grid, make a feature layer, clip with the selected square, and then add all the features back into one feature class. Seems complicated for what seems so simple a task.

Suggestions?