Context. An important number of multiwavelength studies of low ionization nuclear emission-line regions (LINERs) were dedicated to investigate the excitation mechanism responsible for the detected emission lines. Radiative emission from accretion into a super massive black hole (SMBH) is now believed to be, in an increasing number of LINERs, the source of excitation. However, the accretion mode is not yet firmly understood, and could be explained in either a thin accretion disk or a radiatively inefficient accretion flow (RIAF). Aims: Our purpose is to study the X-ray properties of LINER sources with definite detection of a broad Hα emission line in their optical spectra, LINER 1s from Ho et al. sample. These objects preferentially harbor a low luminosity active nucleus at the center and show small or no intrinsic absorption (≤1022 cm-1). We compare their X-ray properties to both X-ray binaries and luminous AGN. Methods: We analyzed all available X-ray archived XMM-Newton and Chandra observations of 13 LINER 1s satisfying the above criterion in a systematic homogeneous way. We looked for any correlations between the X-ray properties and the intrinsic parameters of our sample of LINER 1s. Results: An absorbed power-law gave a good fit to the spectra of 9 out of the 13 sources in our sample. A combination of a thermal component and an absorbed power-law were required in the remaining 4 sources. We found a photon index for our sample between 1.3 ± 0.2 for the hardest source and 2.4+0.2-0.3 for the softest one with a mean value of 1.9 ± 0.2 and a dispersion σ = 0.3. The thermal component had a mean temperature kT ≈ 0.6 keV. Significant short (hours to days) time-scale variability is not common in the present sample and was observed in only 2 sources (NGC 3226 and NGC 4278). Three other sources indicate a possible variability with a low K-S test probability (2-4%) that the nuclear emission originates from a constant source. On the other hand, significant variability on a longer time-scale (months to years) is detected in 7 out of the 9 sources observed more than once. No significant Fe Kα emission line at 6.4 keV was detected and upper limits were derived for the 4 sources with a high enough signal to noise ratio around 6 keV. Finally, we established, for the first time for a sample of LINER 1s, that the photon index Γ is significantly anticorrelated to L2-10 keV/LEdd. Whereas this anticorrelation is in contrast to the positive correlation found for type 1 AGN, it is similar to the one seen in XRBs in their low/hard state where a radiatively inefficient accretion flow is thought to be responsible for the X-ray emitted energy. Figures 1-4, 6, 7, 10 and Appendices A and B are available in electronic form at http://www.aanda.org