diff --git a/NEWS.md b/NEWS.md
index 21ba02f96c..bbcad04707 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -234,6 +234,8 @@
 
 30. `groupingsets` functions now properly handle alone special symbols when using an empty set to group by, [#3653](https://github.com/Rdatatable/data.table/issues/3653). Thanks to @Henrik-P for the report.
 
+31. A `data.table` created using `setDT()` on a `data.frame` containing identical columns referencing each other would cause `setkey()` to return incorrect results, [#3496](https://github.com/Rdatatable/data.table/issues/3496) and [#3766](https://github.com/Rdatatable/data.table/issues/3766). Thanks @kirillmayantsev and @alex46015 for reporting, and @jaapwalhout and @Atrebas for helping to debug and isolate the issue.
+
 #### NOTES
 
 1. `rbindlist`'s `use.names="check"` now emits its message for automatic column names (`"V[0-9]+"`) too, [#3484](https://github.com/Rdatatable/data.table/pull/3484). See news item 5 of v1.12.2 below.
diff --git a/R/data.table.R b/R/data.table.R
index 47c6729320..3885eef2ed 100644
--- a/R/data.table.R
+++ b/R/data.table.R
@@ -1274,9 +1274,9 @@ replace_order = function(isub, verbose, env) {
         if (jcpy) jval = copy(jval)
       } else if (address(jval) == address(SDenv$.SD)) {
         jval = copy(jval)
-      } else if ( length(jcpy <- which(vapply_1c(jval, address) %in% vapply_1c(SDenv, address))) ) {
+      } else if ( length(jcpy <- which(vapply_1c(jval, address) %chin% vapply_1c(SDenv, address))) ) {
         for (jidx in jcpy) jval[[jidx]] = copy(jval[[jidx]])
-      } else if (is.call(jsub) && jsub[[1L]] == "get" && is.list(jval)) {
+      } else if (is.call(jsub) && jsub[[1L]] == "get") {
         jval = copy(jval) # fix for #1212
       }
     } else {
@@ -1285,6 +1285,7 @@ replace_order = function(isub, verbose, env) {
         if (!truelength(jval)) alloc.col(jval)
       }
     }
+
     if (!is.null(lhs)) {
       # TODO?: use set() here now that it can add new columns. Then remove newnames and alloc logic above.
       .Call(Cassign,x,irows,cols,newnames,jval)
diff --git a/R/setkey.R b/R/setkey.R
index fbf8accbe8..0ffdf23a46 100644
--- a/R/setkey.R
+++ b/R/setkey.R
@@ -41,6 +41,12 @@ setkeyv = function(x, cols, verbose=getOption("datatable.verbose"), physical=TRU
     setattr(x,"index",NULL)  # setkey(DT,NULL) also clears secondary keys. setindex(DT,NULL) just clears secondary keys.
     return(invisible(x))
   }
+  if (!missing(verbose)) {
+    stopifnot(isTRUEorFALSE(verbose))
+    # set the global verbose option because that is fetched from C code without having to pass it through
+    oldverbose = options(datatable.verbose=verbose)
+    on.exit(options(oldverbose))
+  }
   if (!is.data.table(x)) stop("x is not a data.table")
   if (!is.character(cols)) stop("cols is not a character vector. Please see further information in ?setkey.")
   if (physical && identical(attr(x, ".data.table.locked", exact=TRUE),TRUE)) stop("Setting a physical key on .SD is reserved for possible future use; to modify the original data's order by group. Try setindex() instead. Or, set*(copy(.SD)) as a (slow) last resort.")
@@ -102,12 +108,9 @@ setkeyv = function(x, cols, verbose=getOption("datatable.verbose"), physical=TRU
   }
   setattr(x,"index",NULL)   # TO DO: reorder existing indexes likely faster than rebuilding again. Allow optionally. Simpler for now to clear.
   if (length(o)) {
-    if (verbose) {
-      tt = suppressMessages(system.time(.Call(Creorder,x,o)))
-      cat("reorder took", tt["user.self"]+tt["sys.self"], "sec\n")
-    } else {
-      .Call(Creorder,x,o)
-    }
+    if (verbose) { last.started.at = proc.time() }
+    .Call(Creorder,x,o)
+    if (verbose) { cat("reorder took", timetaken(last.started.at), "\n"); flush.console() }
   } else {
     if (verbose) cat("x is already ordered by these columns, no need to call reorder\n")
   } # else empty integer() from forderv means x is already ordered by those cols, nothing to do.
diff --git a/inst/tests/tests.Rraw b/inst/tests/tests.Rraw
index a84042353d..de3fe0282e 100644
--- a/inst/tests/tests.Rraw
+++ b/inst/tests/tests.Rraw
@@ -15771,6 +15771,22 @@ test(2086.04, DT[ , sum(a), keyby = list()], data.table(V1=55L))
 test(2086.05, DT[ , sum(a), by = character()], data.table(V1=55L))
 test(2086.06, DT[ , sum(a), keyby = character()], data.table(V1=55L))
 
+# simple queries can create tables with columns sharing the same address, #3766
+x = data.table(a=1L, b=c(1L, 4L, 2L, 3L), c=4:1)
+test(2087.1, x[a == 1L, .(b, b2=b)][ , identical(address(b), address(b2))])
+# setkey detects and copies shared address columns, #3496
+x = data.frame(a=paste0(2:1), stringsAsFactors=FALSE)
+x$b = x$a
+setDT(x)
+test(2087.2, setkey(x, a, verbose=TRUE), data.table(a=paste0(1:2), b=paste0(1:2), key="a"),
+             output='Found and copied 1 column with a shared memory address')
+x = data.frame(a=paste0(2:1), stringsAsFactors=FALSE)
+x$b = x$a
+x$c = x$a
+setDT(x)
+test(2087.3, setkey(x, a, verbose=TRUE), data.table(a=paste0(1:2), b=paste0(1:2), c=paste0(1:2), key="a"),
+             output='Found and copied 2 columns with a shared memory address')
+
 
 ###################################
 #  Add new tests above this line  #
diff --git a/src/data.table.h b/src/data.table.h
index 183d90bee8..469b9e76e7 100644
--- a/src/data.table.h
+++ b/src/data.table.h
@@ -216,4 +216,5 @@ void coerceFill(SEXP fill, double *dfill, int32_t *ifill, int64_t *i64fill);
 SEXP coerceFillR(SEXP fill);
 bool INHERITS(SEXP x, SEXP char_);
 bool Rinherits(SEXP x, SEXP char_);
+void copySharedColumns(SEXP x);
 
diff --git a/src/reorder.c b/src/reorder.c
index e65a58f26c..ec3824efa0 100644
--- a/src/reorder.c
+++ b/src/reorder.c
@@ -21,6 +21,7 @@ SEXP reorder(SEXP x, SEXP order)
         maxSize=SIZEOF(v);
       if (ALTREP(v)) SET_VECTOR_ELT(x,i,duplicate(v));  // expand compact vector in place ready for reordering by reference
     }
+    copySharedColumns(x); // otherwise two columns which point to the same vector would be reordered and then re-reordered, issues linked in PR#3768
   } else {
     if (SIZEOF(x)!=4 && SIZEOF(x)!=8 && SIZEOF(x)!=16)
       error("reorder accepts vectors but this non-VECSXP is type '%s' which isn't yet supported (SIZEOF=%d)", type2char(TYPEOF(x)), SIZEOF(x));
@@ -34,7 +35,6 @@ SEXP reorder(SEXP x, SEXP order)
   int nprotect = 0;
   if (ALTREP(order)) { order=PROTECT(duplicate(order)); nprotect++; }  // TODO: how to fetch range of ALTREP compact vector
 
-
   const int *restrict idx = INTEGER(order);
   int i=0;
   while (i<nrow && idx[i] == i+1) i++;
diff --git a/src/utils.c b/src/utils.c
index f17535c318..a369a8f7bc 100644
--- a/src/utils.c
+++ b/src/utils.c
@@ -173,3 +173,39 @@ bool Rinherits(SEXP x, SEXP char_) {
  return ans;
 }
 
+void copySharedColumns(SEXP x) {
+  const int ncol = length(x);
+  if (!isNewList(x) || ncol==1) return;
+  bool *shared = (bool *)R_alloc(ncol, sizeof(bool)); // on R heap in case duplicate() fails
+  int *savetl = (int *)R_alloc(ncol, sizeof(int));  // on R heap for convenience but could be a calloc
+  int nShared = 0;
+  const SEXP *xp = VECTOR_PTR(x);
+  for (int i=0; i<ncol; ++i) {
+    SEXP thiscol = xp[i];
+    const int thistl = TRUELENGTH(thiscol);
+    if (thistl<0) {
+      shared[i] = true;
+      nShared++;
+      // do not duplicate() here as the duplicate() might fail. Careful to restore tl first to all columns.
+      // Aside: thistl is which column shares the same address as this one in case that's ever useful in future.
+    } else {
+      shared[i] = false;
+      savetl[i] = thistl;  // these are vectors which are all expected to have tl, unlike CHARSXP which often don't (savetl() has CHARSXP in mind)
+      SET_TRUELENGTH(thiscol, -i-1);
+    }
+  }
+  // now we know nShared and which ones they are (if any), restore original tl back to all columns
+  for (int i=0; i<ncol; ++i) {
+    if (!shared[i]) SET_TRUELENGTH(VECTOR_ELT(x, i), savetl[i]);
+  }
+  // now that truelength has been restored for all columns, we can finally call duplicate()
+  if (nShared) {
+    for (int i=0; i<ncol; ++i) {
+      if (shared[i])
+        SET_VECTOR_ELT(x, i, duplicate(VECTOR_ELT(x, i)));
+    }
+    if (GetVerbose()) Rprintf("Found and copied %d column%s with a shared memory address\n", nShared, nShared>1?"s":"");
+    // GetVerbose() (slightly expensive call of all options) called here only when needed
+  }
+}
+