In the spectral reconstruction (SR) problem, reflectance and/or radiance spectra are recovered from RGB images. Most of the prior art only attempts to solve this problem for fixed exposure conditions, and this limits the usefulness of these approaches (they can work inside the lab but not in the real world). In this paper, we seek methods that work well even when exposure is unknown or varies across an image, namely 'exposure invariance'. We begin by re-examining three main approaches - regression, sparse coding and Deep Neural Networks (DNN) - from a varying exposure viewpoint. All three of these approaches are predominantly implemented assuming a fixed capturing condition. However, the leading sparse coding approach (which is almost the best approach overall) is shown to be exposure-invariant, and this teaches that exposure invariance need not come at the cost of poorer overall performance. This result in turn encouraged us to revisit the regression approach. Remarkably, we show that a very simple root-polynomial regression model - which by construction is exposure-invariant - provides competitive performance without any of the complexity inherent in sparse coding or DNNs.