Which way to run the grain

While pattern routing a piece of pine to use as a prototype for a butterfly shapped jewelry box, I ran the grain vertically thinking it would give me the most strength. However, during routing, I am experiencing chunks coming off in the grain direction, esp where I curve in and back out where the side wings meet.
Should I be running the grain horizontally as a general rule? The actual piece will be made of purpleheart, which is orders of magnitude harder than the pine.
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