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 \sim 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 \citet salo2016 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 } \gtrsim 0.5 for barlenses and a/r _ { bar } \lesssim 0.5 for X-shapes , can be explained by projection effects .
Observations at various inclinations are consistent with intrinsic a _ { bl } \approx a _ { X } \approx 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 ^ { \circ } - 60 ^ { \circ } their frequency is \sim 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 with B _ { 4 } being positive at low inclination , and getting negative values for i \gtrsim 40 ^ { \circ } -60 ^ { \circ } , thus implying boxy isophotes .
In total , only about one quarter of barlenses ( with i \leq 60 ^ { \circ } ) show boxy isophotes .

Conclusions : Our analysis are consistent with the idea that barlenses and X-shape features are physically the same phenomenon .
However , which of the two features is observed in a galaxy depends , not only on galaxy inclination , but also on its central flux concentration .
The observed nearly round face-on barlens morphology is expected when at least a few percents of the disk mass is in a central component , within a region much smaller than the size of the barlens itself .
Barlenses participate to secular evolution of galaxies , and might even act as a transition phase between barred and unbarred galaxies .
We also discuss that the large range of stellar population ages obtained for the photometric bulges in the literature , are consistent with our interpretation .