All Values Represented

Question

How many distinct 6-sided dice with values in the set $[6] = \{1,2,\dots, 6\}$ are there such that the following property holds: When this die is rolled twice, there is positive probability that the sum of the upfaces is $k$ for each $2 \leq k \leq 12$. Two dice are considered indistinguishable if the number of faces that show value $i$ for $1 \leq i \leq 6$ is the same for all $i$ between the two dice, regardless of orientation. Assume that each of the $6$ sides appear with equal probability.