Diffracted vortex beams carrying stable tilted total orbital angular momentum (OAM) vector with respect to the propagation direction are generated, when a linearly polarized Laguerre–Gaussian laser pulse impinges on a solid plasma grating target as shown in the particle-in-cell simulation. The diffracted beams are space-separated in the corresponding diffraction directions. It is found that the transverse OAM results from the rotation of the transverse momentum of the vortex harmonics, and can be modulated by the grating period. The stable average transverse OAM provides us deeper insight to the vortex property of light and may provide an additional controllable parameter for potential applications.