  We study the bispectrality of Laguerre type polynomials, which are defined by
taking suitable linear combinations of a fixed number of consecutive Laguerre
polynomials. These Laguerre type polynomials are eigenfunctions of higher-order
differential operators and include, as particular cases, the Krall-Laguerre
polynomials. As the main results, we prove that these Laguerre type polynomials
always satisfy higher-order recurrence relations (i.e., they are bispectral).
We also prove that the Krall-Laguerre families are the only polynomials which
are orthogonal with respect to a measure on the real line.
