Is spinning a new paint can the same as mixing?
Normally the paint store has a shaker not a spinner. But this may be a difference in terminology. 

The paint shaker usually thoroughly stakes the paint to mix it up. They always recommend you buy plenty of paint and mix it all together in 5 gallon buckets so the color is uniform. 

Construction job spec's require painting break to break so floor to ceiling corner to corner to reduce color variations. So, you may have to paint an entire wall to get uniform color.