In an experiment to find focal length of a concave mirror, first we had to estimate its rough focal length. I kept an object at point which was beyond the focus in front of the mirror and got its real image on the screen. But as soon as I removed the screen, I could even see an inverted image of the needle behind the mirror (i.e., virtual image). How is this possible as the concave mirror always form a real image for an object kept at a distance greater than the focal length in front of it?
to see the real inverted image "behind" the mirror is kind of an optical illusion. The picture is still in front of the mirror, but your eyes can not find a frame of reference, so it "sees" the picture at the usual place you see it in plane mirror, a virtual picture would not be inverted. to help your ey to see the picture, at the place where it really is, place som kind of mauve frame instead of the screen, sometimes it even helps if you put just your finger beside the place the screen was before and you see the real picture in the air beside your finger.
When I performed the experiment I saw an inverted image which appeared virtual (i.e not possible).
First of all, it must be understood that our eyes cannot distinguish real images from virtual images directly. Usually we define real images as those which could be formed on a screen whereas virtual images cannot be formed on a screen. When you look yourselves in a plane mirror, virtual image is formed behind this mirror. But you are able to see the virtual image because your eye lens forms a real image of the virtual image formed by the plane mirror on the retina (sounds complicated but it's really easy to understand).
Suppose you are standing in front of a concave mirror. Irrespective of your distance from the mirror you'd see something inside the mirror. When you are within the focal length of the mirror, you'd see an erect magnified image of your face. This is the principle behind shaving mirrors. I hope you know to verify this using ray diagrams.
Coming to your question:
The concave mirror forms a real inverted image at its focal plane when the object is at infinity (practically large finite distances compared to the focal length will do). When your eyes are within the focal length you might see a blurred inverted image. Even when your eyes are exactly at the focal length, you'd still see a blurred image. As you move away gradually, you'd see a clear inverted image of the distant object.
An obvious question arises in the reader's mind: why do we see a blurred image at shorter distances but see a clear/sharp image at comparatively larger distances? This is something to do with the experimenter's least distance of distinct vision (LDDV). You may consider the real image formed by the mirror as an object for your eyes. When the distance from the focal plane and your eyes is less than your LDDV your eye lens can't focus it properly on the retina. When it is larger than the LDDV you'd see a clear image of the object (which is in fact the real inverted image of the distant object formed by the concave mirror).
You are actually seeing an imaginary object floating in air which looks as if it's behind the mirror, an optical illusion as described in this answer due to lack of a reference. Suppose your LDDV is very (very) large then you'd not be able to see a clear image beyond the focal length at comparatively shorter distances.