Appts. for pre-calcining and/or preheating cement material comprises (a) a flow system which allows flow of off-gas from the rotary kiln (12) to the calciner for the preheated material and which is located between fixed suspended preheaters (11a-e) and the kiln (12); (b) a flow system (15) which allows oxygen-rich combustion air to flow from the clinker cooler (14) to the calciner for combustion of supplied fuel; and (c) an arrangement for delivering the pre-calcined material into the lowermost part of the preheater. The novelty is that the branches of the combustion air supply system (15) from the clinker cooler (14) are connected to only one side of the pre-calciner (10) in a tangential manner so that the combustion air, delivered by the system (15), can calcine the supplied material while forming a unidirectional double rotating air stream.