Symmetric matrix for cluster analysis

#1
Does a distance matrix used in a cluster analysis need to be symmetrical? My past experience with cluster analysis usually involved a lower triangular distance matrix but a recent method I came across for a study I am developing results in a symmetrical matrix. A search of the internet does not provide much consensus on the topic. The study is on ecological niche overlap and the method I am thinking of using is here in a small R-based reproducible format:
##### play with rasters ##### ##
build rasters r1 <- raster(nr=18, nc=36)
r2 <- raster(nr=18, nc=36)
r3 <- raster(nr=18, nc=36)
## populate rasters with random values
set.seed(0) r1[] <- runif(ncell(r1))
r2[] <- runif(ncell(r1))
r3[] <- runif(ncell(r1))

## checking: visualize these randomly generated raster files
plot(r1)
plot(r2)
plot(r3)
## checking: explore the values within the raster
class(r1)
summary(r1)
str(r1)
ncells(r1)
## create a vector of the raster names rasters
<-list(r1,r2,r3)
## create an empty vector for use in a for loop
vec <- c()
library(dismo)
for(i in seq_along(rasters)) {
ov <- nicheOverlap( rasters[[1]], rasters[], stat = "I", mask = FALSE, checkNegatives = FALSE
)
vec <- c(vec, ov)
}