An obvious explanation is that the DNA in the lyonized X chromosome is methylated, where as the DNA in the active, X chromosome is not.
To check this hypothesis Peter Jones and Lawrence Shapiro grew cells in the presence of drug 5-azacytosine, which prevents DNA methylation.
This reactivated the lyonized the X chromosome.
Furthermore, Shapiro showed these reactivated chromosomes could be transferred to other cells and still remain active.
Grewal and Moazed (2003) “Heterochromatin and epigenetic control of gene expression” Science 301:798
Goldmit and Bergman (2004) “Monoallelic gene expression: a repertoire of recurrent themes” Immunol Rev 200:197