The map above shows eastern Cook and Will Counties just south of Chicago, Illinois. You can see that the section line mile grid has been imposed on the landscape and creates an efficient grid of Streets. The Interstates were added later.
There were of course Native American trails first that formed the initial roads. Sauk Trail and Monee Road on the map below are examples. Portions of these roads have been brought into the grid but some portions of the roads maintain their original route.
The map shows Park Forest, Illinois, a planned new town developed after World War II. It is a nice community and I lived there for a few years. Sauk Trail and Monee Road were Native Amercan trails but the curvilinear roads added afterwards were designed by the planners. They meandered about in a confusing manner. South Orchard Drive, for example, began to the north as a major entrance to the town. If you follow it south you can see that it turns into a local street that crosses Sauk Trail and then loops back to Sauk Trail. This confused everyone.
I think it is best that if a road starts off on an east west alignment that it stays on that alignment where possible.
Another example is shown below. The longest and most important road in this development is called Creekside Trail, a good name since it follows the creek along the northern portion of the development. This development is in Sarasota, Florida.
But the road is named W. Country Club Lane where it enters from the collector road, Whitfield Ave. This comes from an earlier adjacent subdivision. It then changes in Creekside Trail, heads east, then forms a large circle and returns to W. Country Club Lane. Confuses everyone.