New CCD photometry has been obtained for the RR Lyrae variable star XZ Cygni. An analysis of old and new photometry confirms earlier results that XZ Cyg exhibits the Blazhko effect and that its Blazhko period has changed over time. These changes in the Blazhko period are anticorrelated with observed changes in the primary period of XZ Cyg. During the first half of the 20th century, XZ Cyg had a Blazhko period of approximately 57.4 days. Beginning in 1965, its primary period underwent a steep decline in several steps. Coincidentally, its Blazhko period increased to about 58.5 days. In 1979, the primary period suddenly increased again. After an interval in which the Blazhko effect was small, the Blazhko effect reestablished itself, with a period of approximately 57.5 days. When its Blazhko period is near 57.5 days, XZ Cyg has also shown a tertiary period of 41.6 days. We confirm that there is evidence for a longer 3540 day period in photometry obtained during the first half of the 20th century. XZ Cyg is compared with three other RR Lyrae stars that also appear to show changing Blazhko periods. The observed changes in the length of the Blazhko period of XZ Cyg constrain possible explanations for the Blazhko effect. In particular, they argue against any theoretical explanation that requires that the Blazhko period be exactly equal, or directly proportional, to the rotation period of the star.