Boxy/Peanut/X-shaped (B/P/X) bulges are studied using the 3.6

âµm images from the Spitzer Survey of Stellar Structure in Galaxies,

and the Ks -band images from the Near-IR S0 galaxy Survey. They are

compared with the properties of barlenses, defined as lens-like

structures embedded in bars, with sizes of \sim 50 per cent of bars

and axial ratios of \sim 0.6 - 0.9. Based on observations (extending

Laurikainen et al.) and recent simulation models, we show evidence

that barlenses are the more face-on counterparts of B/P/X-shaped

bulges. Using unsharp masks 18 new X-shaped structures are identified,

covering a large range of galaxy inclinations. The similar masses and

red B - 3.6 µm colours of the host galaxies, and the fact that the

combined axial ratio distribution of the host galaxy discs is flat,

support the interpretation that barlenses and X-shapes are physically

the same phenomenon. In Hubble types -3 < T < 2 even half of the bars

contain either a barlens or an X-shaped structure. Our detailed 2D

multicomponent decompositions for 29 galaxies, fitting the

barlens/X-shape with a separate component, indicate very small or

non-existent classical bulges. Taking into account that the structures

we study have similar host galaxy masses as the Milky Way (MW), our

results imply that MW mass galaxies with no significant classical

bulges are common in the nearby Universe.