There has been some recorded research in this area from more than 150 years back (1858). The concept is called Bilateral transfer of learning.
"Bilateral transfer is an aspect of the transfer of learning and is the transfer of learning or performance from one side of the body after training to the other. So, for example, after training a task with the right hand there, is an improvement in left handed performance."
I could also find a mention of this very specific example (Ref: Slide #8 in this slideshare presentation)-
"He (E H Weber) observed that some children trained to write with the right hand were able without further training to produce very good mirror writing with the left hand".
Why closing the eyes helps may be due to the fact that it reduces distraction but I haven't been able to find any reference for this.