Data analysis often requires methods that are invariant with respect to specific transformations, such as rotations in case of images or shifts in case of images and time series. While principal component analysis (PCA) is a widely-used dimension reduction technique, it lacks robustness with respect to these transformations. Modern alternatives, such as autoencoders, can be invariant with respect to specific transformations but are generally not interpretable. We introduce Transform-Invariant Functional PCA (TI-FPCA) as an effective and interpretable alternative to PCA and autoencoders for functional data. We propose to sequentially approximate the components and show that TI-FPCA outperforms alternative methods in experiments based on synthetic and real data.