Homomorphic encryption (HE), which allows computations on encrypted data, is an enabling technology for confidential cloud computing. One notable example is privacy-preserving Prediction-as-a-Service (PaaS), where machine-learning predictions are computed on encrypted data. However, developing HE-based solutions for encrypted PaaS is a tedious task which requires a careful design that predominantly depends on the deployment scenario and on leveraging the characteristics of modern HE schemes. Prior works on privacy-preserving PaaS focus solely on protecting the confidentiality of the client data uploaded to a remote model provider, e.g., a cloud offering a prediction API, and assume (or take advantage of the fact) that the model is held in plaintext. Furthermore, their aim is to either minimize the latency of the service by processing one sample at a time, or to maximize the number of samples processed per second, while processing a fixed (large) number of samples. In this work, we present slytHErin, an agile framework that enables privacy-preserving PaaS beyond the application scenarios considered in prior works. Thanks to its hybrid design leveraging HE and its multiparty variant (MHE), slytHErin enables novel PaaS scenarios by encrypting the data, the model or both. Moreover, slytHErin features a flexible input data packing approach that allows processing a batch of an arbitrary number of samples, and several computation optimizations that are model-and-setting-agnostic. slytHErin is implemented in Go and it allows end-users to perform encrypted PaaS on custom deep learning models comprising fully-connected, convolutional, and pooling layers, in a few lines of code and without having to worry about the cumbersome implementation and optimization concerns inherent to HE.