Let's analyze both statements.
Statement I: Unpolluted rainwater is naturally acidic. This is because carbon dioxide (CO$_2$) from the atmosphere dissolves in the water to form carbonic acid (H$_2$CO$_3$), a weak acid.
CO$_2$(g) + H$_2$O(l) $\rightleftharpoons$ H$_2$CO$_3$(aq)
This equilibrium results in a pH of approximately 5.6 for normal rainwater. So, Statement I is true.
Statement II: Acid rain is defined as rain that has become more acidic than normal due to atmospheric pollutants. The benchmark for this is the natural acidity of rainwater. Therefore, any precipitation with a pH value less than 5.6 is considered acid rain. The increased acidity is primarily caused by the presence of sulfur oxides (SO$_x$) and nitrogen oxides (NO$_x$) which form sulfuric acid (H$_2$SO$_4$) and nitric acid (HNO$_3$). So, Statement II is true.
Since both statements are true, the correct option is (A).