It is shown that the spinor field equation\(\gamma _a \frac{\partial }{{\partial x_a }}\psi \left( x \right) = 0\), a=1, 2, ..., 2n, inE2n is both covariant for the inhomogeneous orthogonal group of transformationsIO2n and for the groupSO2n+1,1 inE2n+1,1 (or the conformal group of transformations inE2n); the former covariance is named a «natural» covariance, since {ψ(x)} is a carrier space for a faithful, irreducible representation of the groupO2n for allx, but the latter one (or conformal) is only a «restricted» covariance, since {ψ(x)} may not be a carrier space for a faithful representation of the group; in fact, forx=0 onlySO2n andSO1,1 (dilatations) may be nontrivially represented. This restricted covariance is manifested by a theorem which establishes the equivalence (existence of reversible transformations) of the equation above, inE2n, with the system of equations\(\Gamma _A \eta ^A \Gamma _B \frac{\partial }{{\partial \eta _B }}\left( {\Gamma _{2n + 1} + \Gamma _{2n + 2} } \right)\xi \left( \eta \right) = 0\), ΓAηAξ(η)=0,A, B=1, 2, ..., 2n, 2n+1,2n+2, inE2n+1,1. As an example theIO2 naturally covariant spinor field equation in the Euclidean plane (y1,y2)\(\left( {\sigma _1 \frac{\partial }{{\partial y_1 }} + \sigma _2 \frac{\partial }{{\partial y_2 }}} \right)\varphi \left( y \right) = 0\) is shown to be alsoSO3,1 or Lorentz covariant (but restricted covariant) and, according to the above-mentioned theorem, equivalent to the system of equations\(\gamma _\mu x^\mu \gamma _v \frac{\partial }{{\partial x_v }}\left( {\gamma _0 + \gamma _8 } \right)\psi \left( {x = 0} \right)\),xμγμψ(x)=0, μ,v=0, 1, 2, 3, in Minkowski spaceM3,1 and the two-component spinor field ψ(y) above is obtained from\(\frac{1}{2}\left( {1 + \gamma _0 \gamma _8 } \right)\psi \left( x \right)\) and consequently isnot a Weyl spinor field. In a parallel way the massless Dirac equation\(\gamma _\mu \frac{\partial }{{\partial x_\mu }}\psi \left( x \right) = 0\), naturally Poincare covariant in Minkowski spaceM3,1, equivalent to the system\(\Gamma _a \eta ^a \Gamma _b \frac{\partial }{{\partial \eta _b }}\left( {\Gamma _5 + \Gamma _6 } \right)\xi \left( \eta \right) = 0\), Γaηaξ(η)=0,a, b=0, 1, 2, 3, 5, 6, inM4,2, is only «restricted» covariant for the conformal group or forSO4,2 (alternatively the Weyl equation is equivalent to the system above in which the spinor field ξ is substituted by a\(\frac{1}{2}\left( {1 \pm \Gamma _7 } \right)\xi \)semi-spinor or twistor field). For the study of «natural» conformal covariance of spinor fields the equation\(\Gamma _a \frac{\partial }{{\partial \eta _a }}\xi \left( \eta \right) = 0\),a=0, 1, 2, 3, 5, 6, is proposed and studied in the second (forthcoming) part of this work.