Clifford-Steerable Convolutional Neural Networks (CS-CNNs) are a novel class of $ \mathrm{E}(p,q) $-equivariant CNNs that process multivector fields on pseudo-Euclidean spaces $ \mathbb{R}^{p,q} $. These networks are designed to respect the symmetries of these spaces, such as $ \mathrm{E}(3) $-equivariance on $ \mathbb{R}^3 $ and Poincaré-equivariance on Minkowski spacetime $ \mathbb{R}^{1,3} $. CS-CNNs achieve this by using an implicit parametrization of $ \mathrm{O}(p,q) $-steerable kernels via Clifford group equivariant neural networks. This approach allows CS-CNNs to significantly outperform baseline methods in tasks such as fluid dynamics and relativistic electrodynamics forecasting. The key idea is to implement $ \mathrm{O}(p,q) $-steerable kernels implicitly through $ \mathrm{O}(p,q) $-equivariant neural networks for multivectors. This is done by leveraging the connection between the Clifford algebra and the pseudo-orthogonal group. The resulting $ \mathrm{E}(p,q) $-equivariant CNNs are evaluated on various PDE simulation tasks, where they consistently outperform strong baselines, including conventional steerable CNNs and non-equivariant Clifford CNNs. CS-CNNs are particularly effective in tasks involving multivector fields, such as fluid dynamics on $ \mathbb{R}^2 $ and relativistic electrodynamics on $ \mathbb{R}^3 $ and $ \mathbb{R}^{1,2} $. They are the first models to respect the full spacetime symmetries of these problems. The models are also shown to perform on par with analytical solutions for $ \mathrm{E}(2) $ symmetries. The main contributions of this work include the development of CS-CNNs that process full multivector fields on pseudo-Euclidean spaces, the investigation of the representation theory of $ \mathrm{O}(p,q) $-steerable kernels for multivector fields, and the evaluation of $ \mathrm{E}(p,q) $-equivariant CNNs on various PDE simulation tasks. The results demonstrate that CS-CNNs are highly effective in tasks involving multivector fields and are significantly more sample-efficient than traditional methods.Clifford-Steerable Convolutional Neural Networks (CS-CNNs) are a novel class of $ \mathrm{E}(p,q) $-equivariant CNNs that process multivector fields on pseudo-Euclidean spaces $ \mathbb{R}^{p,q} $. These networks are designed to respect the symmetries of these spaces, such as $ \mathrm{E}(3) $-equivariance on $ \mathbb{R}^3 $ and Poincaré-equivariance on Minkowski spacetime $ \mathbb{R}^{1,3} $. CS-CNNs achieve this by using an implicit parametrization of $ \mathrm{O}(p,q) $-steerable kernels via Clifford group equivariant neural networks. This approach allows CS-CNNs to significantly outperform baseline methods in tasks such as fluid dynamics and relativistic electrodynamics forecasting. The key idea is to implement $ \mathrm{O}(p,q) $-steerable kernels implicitly through $ \mathrm{O}(p,q) $-equivariant neural networks for multivectors. This is done by leveraging the connection between the Clifford algebra and the pseudo-orthogonal group. The resulting $ \mathrm{E}(p,q) $-equivariant CNNs are evaluated on various PDE simulation tasks, where they consistently outperform strong baselines, including conventional steerable CNNs and non-equivariant Clifford CNNs. CS-CNNs are particularly effective in tasks involving multivector fields, such as fluid dynamics on $ \mathbb{R}^2 $ and relativistic electrodynamics on $ \mathbb{R}^3 $ and $ \mathbb{R}^{1,2} $. They are the first models to respect the full spacetime symmetries of these problems. The models are also shown to perform on par with analytical solutions for $ \mathrm{E}(2) $ symmetries. The main contributions of this work include the development of CS-CNNs that process full multivector fields on pseudo-Euclidean spaces, the investigation of the representation theory of $ \mathrm{O}(p,q) $-steerable kernels for multivector fields, and the evaluation of $ \mathrm{E}(p,q) $-equivariant CNNs on various PDE simulation tasks. The results demonstrate that CS-CNNs are highly effective in tasks involving multivector fields and are significantly more sample-efficient than traditional methods.