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The Casimir force between two identical bodies, although highly dependent on their geometry and structure of boundaries, is always attractive. However, this force can become repulsive if the nature of the two boundaries is different. We analyze from a global perspective the analytic properties of the Casimir energy function in the space of the consistent boundary conditions F for a massless scalar field confined between two homogeneous parallel plates. The analysis allow us to completely characterize the boundary conditions which give rise to attractive and repulsive Casimir forces. In the interface between both regimes there is a very interesting family of boundary conditions which do not generate any type of Casimir force. We also find Casimirless boundary conditions which are invariant under the renormalization group flow. The conformal invariant boundary conditions which do not generate a Casimir force have not yet been exploited in string theory but open new interesting possibilities.