Non-invasive and continuous blood pressure (BP) monitoring is essential for the early prevention of many cardiovascular diseases. Estimating arterial blood pressure (ABP) from photoplethysmography (PPG) has emerged as a promising solution. However, existing deep learning approaches for PPG-to-ABP reconstruction (PAR) encounter certain information loss, impacting the precision of the reconstructed signal. To overcome this limitation, we introduce an invertible neural network for PPG to ABP reconstruction (INN-PAR), which employs a series of invertible blocks to jointly learn the mapping between PPG and its gradient with the ABP signal and its gradient. INN-PAR efficiently captures both forward and inverse mappings simultaneously, thereby preventing information loss. By integrating signal gradients into the learning process, INN-PAR enhances the network's ability to capture essential high-frequency details, leading to more accurate signal reconstruction. Moreover, we propose a multi-scale convolution module (MSCM) within the invertible block, enabling the model to learn features across multiple scales effectively. We have experimented on two benchmark datasets, which show that INN-PAR significantly outperforms the state-of-the-art methods in both waveform reconstruction and BP measurement accuracy. Codes can be found at: https://github.com/soumitra1992/INNPAR-PPG2ABP.