Context. Morphological characteristics of Boxy/Peanut bulges are studied, in particular whether most of the flux associated to bulges in galaxies with masses similar to those of the Milky Way at redshift z∼0, could belong to vertically thick inner part of the bar, in a similar manner as in the Milky Way itself. At high galaxy inclinations such structures manifest as Boxy/Peanut/X-shape features, and near to face-on view as barlenses. We also study the possibility that bulges in some fraction of unbarred galaxies could form in a similar manner as the bulges in barred galaxies. Aims. We use the Spitzer Survey of Stellar Structure in Galaxies (S 4 G) and the Near-IR S0 galaxy Survey (NIRS0S), to compile complete samples of galaxies with barlenses (N = 85), and X-shape features (N = 88). A sample of unbarred galaxies (N = 41) is also selected, based on similarity in their surface brightness profiles with those of barlens galaxies. Sizes and minor-to-major axis ratios (b/a) of these presumably vertically thick inner bar components are compared, and interpreted by means of synthetic images using N-body simulation models. Barlenses and their parent galaxies are divided into different sub-groups. Their possible parent galaxy counterparts in galaxies where the barlenses are manifested as X-shape features, are also identified. Methods. Unsharp mask images are created for all 214 sample galaxies. These images are used to recognize the X-shape features, and to measure their linear sizes, both along and perpendicular to the bar. For detecting possible boxy isophotes (using B 4 -parameter), isophotal analysis is also performed for the barlens galaxies. In the interpretation N-body simulations from are used: the models, exhibiting Boxy/Peanut/X/barlens morphologies, are viewed from isotropically chosen directions, covering the full range of galaxy inclinations in the sky. The created synthetic images are analyzed in a similar manner as the observations. Results. This is the first time that the observed properties of barlenses and X-shape features are directly compared, over a large range of galaxy inclinations. A comparison with the simulation models shows that the differences in their apparent sizes , a/r bar 0.5 for barlenses and a/r bar 0.5 for X-shapes, can be explained by projection effects. Observations at various inclinations are consistent with intrinsic a bl ≈ a X ≈ 0.5 r bar : here intrinsic size means the face-on semimajor axis length for bars and barlenses, and the semilength of X-shape when the bar is viewed exactly edge on. While X-shapes are quite common at intermediate galaxy inclinations (for i = 40 • -60 • their frequency is ∼ half of barlenses), they are seldom observed at smaller inclinations. This is consistent with our simulation models which have a small compact classical bulge producing a steep inner rotation slope, whereas bulgeless shallow rotation curve models predict that X-shapes should be visible even in face-on geometry. The steep rotation curve models are also consistent with the observed trend ...