The physical meaning of the Levi-Civita spacetime, for some "critical" values of the parameter σ, is discussed in the light of gedanken experiments performed with gyroscopes circumventing the axis of symmetry. The fact that σ = 1/2 corresponds to flat space described from the point of view of an accelerated frame of reference, led us to incorporate the C-metric into discussion. The interpretation of φ as an angle coordinate for any value of σ, appears to be at the origin of difficulties.