We present a spectral variability study of the XMM-Newton and Suzaku observations of one of the most extreme narrow-line Seyfert 1 galaxies, IRAS13224-3809. The X-ray spectrum is characterized by two main peculiar features: (i) a strong soft excess with a steep rise below about 1.3 keV and (ii) a deep drop in flux above 8.2 keV. Although absorption-based interpretations may be able to explain these features by a suitable combination of ionization, covering factors, column densities and outflowing velocities, we focus here on a reflection-based interpretation which interprets both features, as well as the large soft excess, in terms of partially ionized reflection off the inner accretion disc. We show that the two peculiar spectral features mentioned above can be reproduced by two relativistic emission lines due to Fe K and Fe L. The lines are produced in the inner accretion disc and independently yield consistent disc parameters. We argue that the high L/K intensity ratio is broadly consistent with expectations from an ionized accretion disc reflection, indicating that they belong to a single ionized reflection component. The spectral shape, X-ray flux, and variability properties are very similar in the XMM-Newton and Suzaku observations, performed about 5 yr apart. The overall X-ray spectrum and variability can be described by a simple two-component model comprising a steep power-law continuum plus its ionized reflection off the inner accretion disc. In this model, a rapidly rotating Kerr black hole and a steep emissivity profile are required to describe the data. The simultaneous detection of broad relativistic Fe L and K lines in IRAS13224-3809 follows that in another extreme NLS1 galaxy, 1H0707-495. Although the data quality for IRAS13224-3809 does not allow us to rule out competing models as in 1H0707-495, we show here that our reflection-based interpretation describes in a self-consistent manner the available data and points towards IRAS13224-3809 being a very close relative of 1H0707-495 in terms of both spectral and variability properties. These results, together with those based on pure broad Fe K detections, are starting to unveil the processes taking place in the immediate vicinity of accreting radiatively efficient black holes.