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The algebraic and differential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">SO</mi><mfenced open="(" close=")"><mn>3</mn></mfenced></mrow></semantics></math></inline-formula>-invariants of spherical harmonics are studied in this work. The fields of rational algebraic and rational differential invariants and their applications for the description of regular <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">SO</mi><mfenced open="(" close=")"><mn>3</mn></mfenced></mrow></semantics></math></inline-formula>-orbits of spherical harmonics are given.